Complex Function Theory for Functions with Arguments and Values in Locally Convex Linear Topological Spaces Citation Bernholtz, Benjamin (1952) Complex Function Theory for Functions with Arguments and Values in Locally Convex Linear Topological Spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/NSNX-7J43. https://resolver.caltech.edu/CaltechTHESIS:10042017-091604604 Abstract In Chapter I a brief introduction to the basic notions of locally convex linear topological spaces is given. In Chapter II, a theory of analytic functions is developed for functions of a complex variable with values in a sequentially complete locally convex complex linear topological space (l.t.s). The theory is sketched of continuous linear functions on one sequentially complete locally convex complex l.t.s. to a second such space. In the same spirit some theorems relating to functions of several complex variables taking their values in a sequentially complete locally convex complex l.t.s. are developed. In Chapter III, functions on one sequentially complete locally convex complex l.t.s to a second such space are studied and in particular notions of differentiability and analyticity. An analogue of the Cauchy-Riemann theory of functions of a complex variable is discussed. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics and Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Michal, Aristotle D. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1952 Record Number: CaltechTHESIS:10042017-091604604 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10042017-091604604 DOI: 10.7907/NSNX-7J43 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10481 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 04 Oct 2017 17:01 Last Modified: 10 May 2023 22:41 Thesis Files Preview PDF - Final Version See Usage Policy. 31MB Repository Staff Only: item control page

COMPLEX FUNCTION THEORY FOR FUNCTIONS WITH ARGUMENTS AND VALUES IN LOCALLY CONVEX LINEAR TOPOLOGICAL SPACES Thesis by Benjamin Bernholtz In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1952 ACKNOWLEDGMENTS The ·author wishes to express his graditude to Professor A. D. Michal for guiding him into the path which led to these researches and to thank him for his constant advice and encouragement while this work was in progress. ABSTRACT In Chapter I a brief introduction to the basic notions of locally convex Jinear topological spaces is given. In Chapter II, a theory of analytic functions is developed for functions of a complex variable with values in a sequentially complete locally convex complex linear topological space (l.t.s). The theory is sketched for the case when the function values lie in the Jinear space of continuous linear functions on one sequentially complete locally convex complex 1. t . s. to a second such space. In the same spirit some theorems relating to functions of several complex variables taking their values in a sequentially complete locally convex complex l.t.s. are developed. In Chapter III, functions on one sequentially complete locally convex complex l.t . s. to a second such space are studied and in particular notions of differentiability and analyticity. An analogue of the Cauchy-Riemann theory of functions of a complex variable is discussed. TABLE OF CONrENTS Page Chapter I. Locally Convex Linear Topological Spaces 1.1 Introduction 1 1.2 Postulates for a Locally Convex 1. T. s. 1 1.3 Pseudo-norms and F-metrics 5 1.4 Examples of Locally Convex L. T. s. 11 1.5 Couple and Composite Locally Convex L. T. s. 14 1.6 Continuous Linear Functionals and Weak Topologies 18 1.7 Multilinear Functions 25 1.8 Extended Locally Convex Topological Linear Rings 28 1.9 Examples of Extended Locally Convex Topological Linear Rings Chapter II. 31 Analytic Functions of a Complex Variable 2.1 Introduction 35 2.2 Continuity and Differentiability 35 2.3 Complex Integration 38 2.4 2.5 Regular Functions and Cauchy's Integral Theorem 45 P~1er Series and Taylor Expansion 54 2.6 Laurent's Expansion and Singularities 64 2. 7 Regular Operator Functions 70 2.8 Vector Functions of Several Complex Variables 73 Chapter III. Functions on Vectors to Vectors 3.1 Introduction 77 3.2 G-Differentiability 77 3.3 Series Expansions of G-differentiable Functions 82 3.4 F-d.ifferentiability and Partial Differentials 87 3.5 Polynomials 92 3.6 The Series Expansion f or Ana~tic Functions 97 3. 7 Further Properties of Analytic Functions 104 3. 8 The Generalized Cauchy-Riemann Equations 108 References 118 -1- Chapter I Locally Convex Linear Topo lo gi cal Spaces 1. 1. Introduction. The main purpose of this chapter is to give a brief introduction to the basic notions in locally conve..~ linear topological spaces. As much of the material is taken from the current literature, proofs will, in general, be omitted or merely sketched. Where proofs ··are included they are intended either to make up a deficit in this respect in the literature or to correspond to results which are more or less new. § 1. 1.2. Definitions and Fundamental Properties Postulates for a Locally Convex Linear TQQQ.logical Sua~ Definition 1 • 2. 1. Let 1 be a non-empty set of elements x, y, z, ••• , and let A denote either the real or complex numbers. 1 will be called a linear system over A, if its elements admit of two operations, addition and scalar multiplication, subject to the following col)ditions. Addition of elements satisfies: A • 1 Every ordered pair of elements x, y of L has a unique sum x + y belongi ng to L. A2. Addition is commutative, i.e., x + y ::::: y + x. A3. Addition is associative, i.e., (x + y) + z =x + (y + z). A4. There is an element 8 in L, called the zero element, such t hat x + 8 ::::: 8 + x = x for all x in L. A • 5 To every x there corresponds an element -x, its negative, such tha.t x + ( -x) = e. -2- Scalar rrmltiplication satisfies: s1 • To every number a in A, and every x in L, there corresponds a uniquely defined scalar product ax in L. s2. (a + b )x =ax +bx, a and b in A, x in L. s3. a(x + y) == ax + ay, a in A, x and y in L. S4• a(bx) == (ab)x, a and b in A, x in L. S5• 1x = x. ' For the elementary properties of linear systems see (1) and (2). If M and N are subsets of L, the symbols -M, M + N, M - N, denote the sets [ -x ~ , JvI and y is in N. set { bx ) {x + y) , [ x - y) respectively where x is in If B is a subset of A, the symbol BM st&'1ds for the where b is in B and x in M. According as A is the real or complex numbers, a linear system 1 will be called a real or a complex linear system. R will always denote the real numbers and K the complex numbers. Definition 1.2.2. be a subset of L. Let L be any complex linear system and let M M is said to be (i) convex, if x,y in M implies ax + (1-a)y is in M, 0 < a < 1, (i. i) _ . 1 ar, ii . ~ f or x in . M . in . 'l\ifl , f or 0 ""'< rAv.; ..:::..- 2 TT , circu , , e i0 x is and (iii) balanced, if for each x f. e, there exists a real number a /::. o, such that ax is in M. In the case of a real linear system, (ii) is replaced by -3- (iv) M is said to be symmetric, if whenever xis in IE, then so is -x. Definition 1. 2.3. A complex linear system 1 i s called a locally convex complex linear topolo gical space (3) if the following hold. (i) 1 is a Hausdorff topolo gical space (2). (ii) There exists a fundamental system of neighbourhoods of denoted by __,e = { C) circular and balanced. = { C(8 )J e, formed of sets which are convex, The neighbourhoods ?1. = [ N(x) J of an arbitrary point x of 1 are obtained by translating the neighbourhoods of the ori gin, N(x) = x + C( 8 ). (iii) In 1 the operations of addition and scalar multiplication are continuous under the given topology, i.e., x + y and (x, y in L, Jx ~ in K) are each continuous jointly in both arguments. The phrase linear topological space(s) will always be abbreviated to 1. t. s. To define a locally convex real 1. t. s. circular is replaced by symmetric and K by R in Definition 1.2.3. Various authors (4,5) have given different sets of postulates for such spaces and a discussion of the re la ti onship among them is to be found in ( 6 ) • In Definition 1. 2. 3 it would have been sufficient to require only that the nei ghbourhoods defining the topology be convex sets but then such a topology can always be replaced by an equivalent topo logy in which the defining nei ghbourhoods are convex, circular and balanced. Theorem 1.2.1. The Hausdorff separation axiom is equivalent to the statement that the intersection of all the neighbourhoods of the element 8 consists only of e. Proof. Suppose the separation axiom holds but in addition to e, some element x f. 8. n c contains' ,£ Then there exist neighbourhoods C of 8 and N(x) of x such that C f'l N(x) = 0, the empty set, giving a contradiction. If n c = e, take x f. e. ,e, such that x is not in c. ~ Then there exists some c in A On the other hand since (l N(x) "Yl. = = x + Q C( 8) = x there is some neighbourhood N(x) of x + C( 9 ) x not containing 8. These last tvm statements together say that L is a T -space, i.e., if x f. y there is a neighbourhood N( x) of x to which 1 y does not belong. This last implies ( 7 ) that 1 not only satisfies the Hausdorf f separation axiom but is indeed regular. Definition 1.2.4. A sequence~' x 2, ••• , xn,••• of elements of a locally convex 1. t. s. is a convergent sequence if subsequently = n 0 (C) may always be assigned so that m > n ~ n (C) implies x - x is in c. o m n to the choice of an arbitrary neighbourhood C of 8, a nu.,~ber n Definition 1.2.5. A sequence 0 x.,, x 2, ••• , xn' ... in a locally convex l.t.s. 1 is convergent to an element x of L, called its limit, if subsequently to the choice of an arbitrary nei ghbourhood C of 8 a numb er n 0 = n0 ( C) may always be as s i gned such that n ~ n 0 ( C) implies x - x lim n-+ oo x n = x. n is in C. As usual, this is expressed by writing -5- Definition 1. 2. 6. A locally convex 1. t. s. is said to be sequentially complete if every convergent sequence in L is convergent to an e lement in L. 1.3. Pseudo-norms and F-metrics. An alternative method of defining a locally convex 1. t. s. will be given here. Definition 1.3.1. A real valued function p defined on a comp lex linear system L is called a complex (triangular) pseudo-norm on L if (i) p(x) ~ o, (ii) p( ~ x) = I t J p(x), where (iii) p(x + y) ~ ~ is in K, and p(x) + p(y). If L is a real linear system (ii) is replaced by (iv) p(ax) = I a I p(x), where a is in R, and p is called a real (triangular) pseudo-norm on L. The notion of a triangular p se"L;.do-norm was ini:,rociuced by J . von NeCLrnann ( 4 ) and generalized by Hyers (8). It differs from a norm only in that the vanisl:1ing of t he pseudo-norm of an element x does not imply x = e. Definition 1.3.2. An arbitrary set D with elements d,e,f, ••• to get her with a binary relation, > , is called a directed system if: (i) Either d ) e or d j e, for every pair, d,e in D. (ii) If d (iii) Given d, e in D, t he re exists f in ~, such that f > e, a nd e > f, then d > f. > d, f > e. -6- Definition 1.J.J. A complex (real) linear system L will b e said to be pseudo-normed with respect to a directed systen D if there exists II x a real valued function II d' defined for all x in L and d in D such that (i) JI x ii dis a complex (real) triangular pseudo-norm, (ii) I/ x ii d (iii) II x II d ? II x II e whenever d > e, for all x in L. = 0 for all din D implies x = 8, and Given a complex (real) linear system L pseudo-normed with respect to a directed system D, the sets = {x in 1 / JI x d < E J , C( 8; d, E ) 11 defined for every d in D and E ::> o, form a fundamental system of convex, circular (symmetric) and balanced nei ghbourhoods of e. The nei ghbourhoods of an arbitrary point x of 1 are the sets N(x ; d, E: ) ::: 0 { x in L J II x - x II 0 all possible values. d <EJ For, if x 1 Then N(x ;d, - ) and N(y ;d, y 0 0 respe cti vel;:r. X O- y 0 11 d > 0 takes on f. y 0 , by Defin:L ti on 1 • 3. 3 there Suppose II x 0 - y 0 II d = 1. j) are disjoint neighbourhoods of x and 0 Since otherwise, x being co1mnon to both, it would follow that 1 = II x + 11 0 II x 0 - y 0 II d f. O. is some d in D for which 3 E That these nei ghbourhoo ds satisfy the Hausdorff separation axiom is clear. 0 as d ranges over D and 0 - y 0 II d = II x 0 - x + x - y 0 11 d ~ II x 0 - x II d <32 ' leading to a contradiction. The continuity of addition is a consequence of (iii) of Definition 1.J . 1 and that of scalar multiplication of (ii) and (iii) of the sa.u e definition. Hence, -7- using the pseudo-norms, a locally convex topolog-:J can be defined in 1 called t he locally convex topology generated by the pseudonorms. Theorem 1.J.1. convex 1. t. s. Every pseudo-normed linear ~JStEfil is a l ocally Conversely, given any locally convex 1. t. s. 1 there exists a directed system D, called the associated directed system, with respect to vvhich 1 may be pseudo-normed in such a way that the pseudo-norms generate a topology equ.i.valent to the given topology in 1 ( 8). Proof. indicat ed. The proof of the first statement has already been As for the second, the directed system mentioned there may be identified with the given syst en of nei ghbourhoods { c(e) } = in fi . • If C is in A the corresponding pseudo-norm is defined as follows: II x II C = inf { a It is clear that > 0 I x is in aC } • II x II C ~ 1 for x in C. That II x II C is actually a triangular pseudo-norm follows from the fact that C is convex, circular ( syrrm1etric) aDd balanced; that II x II C = O for all C implies x = 8 follows from the Hausdorff separation axiom or since c 1 n,e, C = 8. Finally > c2 me a..YJ.s c1 C c2 it is true that II x 11 c ~ II x II c if 1 2 c1 > c2. It is now evident that locally convex 1. t. s. and pseudo-normed - 8- linear systems are identical objects. Hence , in speaking of locally convex l.t.s. instead of discussing questions of topology and analysis in terms of the given nei ghbourhoods, they will be discussed in terms of the pseudo-nonns. The question as to when two systems of pseudo- norms define equivalent topolo gies (in which case the systems are said to be equivalent) t hen arises and is answered by Theorem 1.3. 2. Let L be a locally convex l.t.s. and D, D1 two associated directed systems. In order that the systems of pseudo- norms II x fl d' d in D, and II x ti d', d 1 1 in D , be equivalent it is necessary and sufficient that for each d in D th ere exist numbers M (d) 1 > o, }g 2(d) > 0 and ct' in D' such that for all x in L M1 ( d) ll x II d ~ II x II d 1 ~ M ( d) II x II d 2 • 1 l x j II x II d r ( 1v''1 ( d) E j , N = [ x \ II x II d <EJ and N~ = l x j llx "ct' < M2 (d) E J • It follows from the above Let N1 = Proof. inequalities that N t C 1 N C N 1 f 2 and the topologies are equivalent. Suppose D and D are equivalent. Consider N = l x I II x 11 d < 1 J . = [ x I 11 x II d' < E, E = E (d) J II x 11 d < 1 imp 1i es II x II d, < E • With Then there exists a neighbourhood of e, N' 1 such that N C N , i.e., 0 < I t 0 I < 1 and t 0 real, there is one and only one integer k = k( d) I t 0 I < II t 0 kx II d < for which Thus II t 0 k x II d r < E Put ting M ( d) 2 E and II x II d 1 < - - k = -{- = E t( d) 0 1 as long as x 0 < I= 8, or II x ll d I= O. E - t - II x II d. It 0 I o this gi. ves ll x II d, ~ M ( d) II x ll d. 2 This relation is true for x = 8, and for those x for which II x II d = 0 since -9- = o. Otherwise there would exist a real number a such that II ax II d, = I a I IJ x Hd, > E while II ax II d = I al II x II d = o, a contradiction. this last implies II x II d, In terms of pseudo-norms Definitions 1. 2.4 and 1. 2.5 read as follows. Definition 1.J.4. Let L be a locally convex l.t.s. with A sequence associated directed system D. a convergent sequence if given n E l xn } of elements of L is > o, d in D, there is a number = n o (d, E ) such that m > n ~ n o (d, E ) implies II x m - x n II d < E o Definition 1.J.5. [ x n " With Land Das in Definition 1.J .4, a sequence J in L is convergent to an element x in L, called its limit if given € > o, din D, a number n 0 = n 0 (d, E ) may always be assigned such that n ~ n ( d, o f Definition 1.3.6. ) implies 11 x - x II < f n d • A real valued function defined on a complex 9 ) and denoted by linear systen is called a complex F-metric ( 1, x in L, if l x l ~ 0 and I x I = 0 if and only if x = e, ( i) ' ( l..l ; (iii) (iv) I ~ + x2 1 ~ l ~ I + I x2 1 i0 l e x i = I x I ' 0 ~ 0 ~ 2Tf ' and ' ~ n~ '> ( ~ n' ~ in K) and I x n ~ nx n - ~ x I -i- o. For a real linear system L (iii) is replaced by ... I ) ( lll 1-x I = I x I - x I ~ 0 implies I x I, -10- l~n } and the complex sequence sequence If { an } Ixi 'Ni th a n -+ in (iv) is replaced by a real a. is an F-metric on a linear system 1 a topology can be introduced in 1 using nei ghbourhoods N(8, N( e' E ) c ) of the origin where = l x in 1 I I x l < ~ J Definition 1.3. 7 (1 0). A complex (real) 7' -space is a sequentially complete locally convex complex (real) 1. t. s. in which a complex (real) F-metric can be introduced defining an equivalent topo lo gy. A result due to G. Birk.1-ioff on F-metrics (11) implies that the J: -spaces satisfy the first axiom of countability. Hence the topolo gy of JI ']: -spacescan be defined by a sequence norms such that II x II n ~ ll x II n+1 x II m of pseudo- • The question of how locally convex 1. t. s. differ from normed spaces can be answered by introducing bounded sets. Definition 1.3.8 ( 4 ) . A set S contained in a locally convex l.t.s. with directed system Dis said to be bounded if sup ll x lld <. x in S ,,,, for every- d in D. Equivalent definitions have been given by Ko:L'Tlogoroff (7 ) and Michal and Paxson ( 1 2, 13 ). Theorem 1.3.3 ( 7 ). In order that a norm defining an equivalent topolo gy may b e intro duced into a locally convex 1. t. s. it is necessa:r:ir and sufficient that 1 contain a bouncied open set having the ori gin as an interior point. -11- 1.4. Examples of Locally Convex Linear Topological Spaces. Example 1.1.i.• 1. Let D be a directed set and Bd,d in D, a family of Banach s paces, the norm of an element Bd being denoted by II • II ct• Suppose (i) Be C Bd vmenever e ( ii) if x is in Be and e Set B = (l Bd. > d, e and d in D, and > d t hen II x II d < Il x II e. B is clearly a sequentially complete locally convex D 1. t. s. pseudo-normed 1"1i th respect to the directed s y stem D with pseudo-norms II x ll d defined for all x in Band din D. As particular instances of this construction D may be t aken as the set of real numbers 1 ~ p ~ CX> and t he corresp onding family of .Banach spaces as the LP(o,1) spaces or t he lp spaces. Properties (i) and (ii) follow from t he known properties of LP and lP. Examples of PJ:' -spaces may be constructed by taking for D the set of pos i tive integers with Bk' k = 1, 2, • • • the corresponding co family of Banac h s paces ( 9 ). F'-re tric I x I B = Q1 Bk is an "'.J. -space and the is given by CX> Ix I 1 =L k=1 2K 11 x II k , x in B. 1 + ll x ll k It is easily verif ied t ha t this F-metric defines a topolo gy equiva l ent to that generated by the p seudo- norms II x II k" sequentially complete locally conv ex 1 . t. s. L co tak i ng 1 -n k=1 Lk(O, 1) . A non-normable w ( 14)is ob tained by -12- = 1, 2, •••• be the Banach spa ces of functions defined Let J:J k' k on 0 ~ t ~ 1 having continuous derivatives of order k and with norm g iven by = 11 x II k ~ k 0~t ~ 1 x = x( t) in ,/) k• = n ofJ k• l:J is the non-normable sequentially complete k=1 -space of infinitely differentiable functions defined on the /j Set J. I x ( i) ( t) I , sup o~ i interval 0 ~ t ~ 1 of the real line (1,10 ) . Example 1. 4. 2. Let D be a directed set and family of Banach spaces. by If xe is in Be with e as a member of < 11 x e B x + y = l xd ) by ~ { xd ) > d, then xe also belongs to Bd and Bd may be denoted by xd. II Set B == TT Bd' that is, Condition (ii) then reads B is the Cartesian D product of the spaces elements in B d its norm will be denoted Suppose conditions ( i) and (ii) of example 1 • 4. 1 are II xd II • fulfilled. If xd is in B d' d in D, a B is a linear systan if addition of B d" , say x == { xd } , y + ( Yd } == = { yd } { xd + Yd ~ is defh1ed by , and scalar multiplication {t xd } • The scalar multiplier $:" is taken to be real or complex according as the Banach s paces Bd are real or complex. II xd II The pseudo-norm II x II d of an element x in B is the norm of xd in the space Bd. B is then a sequentially complete locally convex 1. t. s. pseudo-normed V\ri th respect to the directed system D. As in Example 1.4.1, specific instances of this con- struction are obtained by taking for D the set of real numbers -13- 1 ~ p ~ oo and for the corresponding family of Banach spaces the LP(o,1) or the l p spaces. Further exam p les of '7. .J' -spaces are obtained by taking for D the set of positive integers with Bk' k = 1 , 2,. • • the corresponding farnily of Banach spaces ( 9 ) . B = 7T B k k is now the s pace of all { ~ 1 , where Xi,"" is in Bk' and is an "]. -space, the F- sequences metric being given by 00 L 2k 1 11+~llIIxk II 1 Ixi k=1 If Bk' k = 1,2, •••• is the normed linear space of real numbers then the space B is t h e sp ace (s) of all nurnerical sequences (1 ). Examp le 1.4.3 (10). Let G be a locally compact Hausdorff space whi ch is the union of a countable number of compact sets G where n • n+1 the closure of G is contained in t he interior of G n subset of G is then contained in some G • 11 Each compact Let 1 be t h e complex linear syst011 of cont inuous complex valued flmctions defined on G. A non- normabl e sequentially complete locally convex l.t.s. on 1 is defined by the countab le family of pseudo-norms JI x II = sup I x( t) n t in . ,,v I , x = x( t) in L, n = 1 , 2, ••• n Hence 1 is an J. -space. Particular instances of this construction are obtained by taking G to be the complex plane and Gn to be { }. in K Gn to be I ~I ~ n } ; or by taking G to be the real line and l t in R I I t I ~ n } • -14- 1. 5. Couple and Composite Locally Convex Linear Topological Spaces. Given a locally convex real l.t.s., say L(R), it is possible to construct from it other locally convex l.t.s. Of particular interest are two s p aces which will be called the associated coup le and composite spaces and will be denoted by L( K) and L(C) respectively. L(K) is a loc all y convex complex l.t.s. while L(C) is a locally convex real l .t.s. Theorem 1.5.1. directed system D. Let L(R) be a locally convex real l.t.s. with L( K) will denote the class of all pairs z = (x,y) of elements of L(R) -with equality, addition, multiplication by complex nwnbers and pseudo-norms defined as follows for z z 2 = (~, y1 ), 3" = (a,b) complex, and the directed set D: = (x , y ), 2 1 2 ( i) (~ ,y1) = (x2,y 2) if and only if x 1 == x 2 , Y1 = y 2' (ii) (x1'y1) + (x2,y 2) == (x1 + x2' Y1 + Y2)' (iii) (a,b)(~,y ) == (a~ - by , ay 1 1 1 + b~) where a,b are real, and (iv) II (x,y) II d = sup II x cos (/) + y sin 0 II d' each d in D. o ~ 0 $ 2rr Then L( K), the associated couple space, is a locaJ.ly convex complex 1. t. s. pseudo-normed with respect to the directed systera D. If L( R) is sequentially complete then so is L( K). Proof. Addi tion clearly satisfies A , A , AJ of Definition 1.2.1. 1 2 The zero e l ement of L(K) is (-x, -y). (e,e) wh ile the negative of (x,y) is s 2, SJ' s4 are easily checked. ( a , b ) be t wo complex nmnbers. 2 2 Then For example let (a ,b ) and 1 1 -15- ((~ ,b ) 1 + (a ,b ) )(x,y) = (a1 + a 2 , b 1 + b )(x,y) 2 2 2 = ((a 1 + a 2 )x - (b 1 + t 2 )y, ( a 1 + a 2 )y + (b1 + b 2 )x) = (a1 x - b1 y, a1y + b 1 x) + (a 2x - b~, a 2y + b 2x) = (a1 ,b 1 )(x,y) + (a 2,b 2 )(x,y). As for s , 1 (x,y) = (1,0) (x,y) = (x,y). 5 Similarly it is easily verified by a straightforward computation that for each d in D norm. II (x,y) II d is actually a triangular pseudo- Hmvever, instead of doing this it wi.11 be shown by u si ne; the Minkovrnki functional that (iv) gives the natural definition of pseudonorms for L( K). Consider the family ,,i in L(R). = {C) of nei ghbourhoods of the origin A topolo gy for L( K) is most naturally constructed by taking as neighbourhoods of ( 8 , 8 ) the set of all pairs (C,C~ of elements of .,g • These are in a manner of speaking 11 rectan Eular 11 nei ghbourhoods. They can be replaced by nei ghbourhoods of the fo rm S = ( C, C), C in A , the "s quare" nei ghbourhoods, defi!ling an equivalent topolo gy on L(K). Each S is a convex, symmetric a nd balanced nei ghbourhood of ( 8 , 8 ). Dut L( K) is to be a locally convex comp lex 1. t. s. and so the defining nei ghbourhoods for it must be circular. Let W = 0 ~ n0 e i(C· '~ :Jo ~ 27\ Then as S ranges over the square nei ghbourhoods the sets W form a fundamental system of convex, circular and balanced nei ghb ourhoods of ( 8 , 8 ) defining an equivalent Hausdorff topolo gy on L( K). The nei ghbourhoods { W) form a directed set wh en w 1 > W2 is -16- taken to mean w C 1 Let z be in L( K). W• 2 The pseudo-norm of z with respect to H is defined by II z II W = inf [ h > 0 / z is in hW, z = (x,y) } • Then, II z II w,. =inf { h > 0 z E he i0 .s, 0 ~ 0 ~ 2 TT ze-:iJZ· E hS, =inf [ h >0 =inf ( h >0 I (x cos 0 +y sin (/, =inf >0 [ h l 0 ~ 0 ~ 2 rr} 1 , y cos 0 -x sin 0 E (hC, hC ),o~ 0 ~ 2 rr} x cos ~~ + y sin 0 E hC, o ~ 0 ,; 2TT} y cos 0 - x sin 0 E hC = sup [ ll x cos 0 + y sin 0 l/ C' II x cos( Tf + 0) + y sin( lf + 0) II o ~ 0 ~ 2rr C} = sup II x cos 0 + y sin 0 II c· 0 ~ 0 '5: 2 1T The elements C of Since Yi! = { W J n A form a directed set if c1 c2. eir/JS and S = ( C, C) it is clear t hat the directed set 0 ~ 0 ~ 2 TT may be replaced by the directed set II z II C = > c2 means c1 c II (x,y) II C = sup II o ~ r/J f:: 2 rr --4 to obtain x cos 0 + y sin (/_) II C' t hat is, the expression (iv) has been derived naturally from the neighbourhood definition of the topolo gy in L(K). The mapping X-+ (x, 8 ) is an isomorphism between the space L(R) and the linear sub-space of L( K) of elements of the form (x, 8 ). These may t h en be identified a nd z = (x,y) = (x,e) + (0,1)(y,8) written as z = x + iy. The following inequ.ali ties are useful in this connection. -17- II x II d ~ I\ x + iy II d II y II d ~ II x + iy II d ~ II x II d + 11 x + iy 11 d II y II d Let L( R) be sequentially comp lete and supp ose [ xn,yn } convergent sequence in L( K) so that given d in D, n 0 is a > o, there exists t = n 0 (d, E ) such th2.t form > n ~ n 0 (d, t ) it is true that = JI (xm - xn, m y - y n ) II ct < E 11 (xmm ,y ) - (xn,yn ) II ct • The first t wo inecp alities above then imply ll xm -xn ii d <E [ xn } t h at is, x = lim x and and { yn ~ /l ym -yn ll d <E ' are convergent sequence in L(R). n and y = lim y n • Let Then using the third inequality, /l (x-x,y-y) I/ a, ~ // x-x /l d+ Jj y-y /l d n n n n and this last can be made arbitrarily smaJJ. for given d in D by the choice of n. L(K) is thus sequentia lly complete. Theorem 1.S.2. Let L( R) be a locally convex real l.t.s. and L(K) the associated coup le s pace. z Let z = x + iy. Then t h e function = x - iy, called the conjugate of z is a continuous function of z and = z = z. Proof. For any d in D, II z - z 11 0 d = II z - z 0 II d and t h e result follows. Theorem 1.S.J. Let L( R) be a locally convex real l . t.s. with directed systen D. Let L( C) denote the class o.f all pairs (x,y) of elements of L(C) with equality, addition, multiplication by real -18- numbers, and pseudo-nonns defined as follows for the pairs (~ ,y1 ), (x ,y ), the real number a, and the directed set D: 2 2 (i) (4 ,y1 ) = (x 2,y 2 ) i f and only if 4 = x 2, y1 = y 2, (ii) (~,y1) + (x2,y2) (iii) a(x,y) = (4 +x2' Y1 +y2)' = (ax,ay), (iv) II (x,y) II d = II x II d + II y ll d' each d in D. Then L(C), the associated composite space, is a locally convex real 1. t. s. pseudo-normed with respect to the directed system D. If L(R) is sequentially complete then so is L(C). 1. 6 Continuous Linear Functionals and Weak Topologies. Definition 1. 6. 1. 1. t. s. and D and D 1 I Let L and L be two locally convex complex their associa.ted directed systems respectively. Let T(x) be a transformation on L into 1 ( i) 1 • T is said to be linear if for all 4 and x T(~ and for all ~ i f given d in L + x 2 ) = T(4) + T(x 2 ), in K, x in 1 T( ~ x) (ii) 2 = ~ T(x). T is said to be bounded with respect to the pseudo-norms 1 1 in D' there exists din D and M(d,d ) ) 0 such that for all x in 1 II T(x) ll ct' (iii) ~ M(d,d') II x II ct• If 1 1 is the space of complex numbers K, and T satisfies the conditions of (i), then Tis said to be a complex linear functional. -19- If Land L I are locally convex real l.t.s. Definition 1.6.1 (ii) remains as it is while the second statement of Definition 1.6.1 (i) must be changed to read - for all a in R and x in L, T( ax) particular L I = aT(x). If in is the space of real numbers and T satisfies the amended form of Definition 1.6.1 (i) Tis said to be a real linear functional. Theorem 1 • 6. 1 ( 6 : • 1 Let L and L be two locally convex l.t.s. 1 and let D and D be the associated directed systems for L and 1 respectively. A linear transformation T(x) on L into L 1 1 is continuous everywhere if and only i f it is bounded with respect to the pseudonorms. Corollary. Let f(x) be a linear functional defined on a locally convex 1. t. s. L vvith associated directed system D. f(x) is continuous i f and only if there exists d in D and a real number M( d) that I f(x) I ~ M(d) > 0 such II x II d• It is of fundamental importance for later applications to know if there are any continuous linear functionals on a locally convex 1. t. s. and i f so, if there are sufficiently many to distinguish between elements of L. The first question is solved in Theorem 1.6.2 (15). Non-zero continuous linear functionals exist on a l.t.s. if and only if t he space contains an open convex set containing the ori gin but not the whole space. Hence non-zero continuous linear functionals certainly exist on locally convex 1. t. s. As in 3anach spaces the second question may be answered with the aid of -20- Theoren 1.6.J (1). Let L be a real linear system and let p(x) be a positive real valued function defined on L such that p(x + y) ~ p(x) + p(y), p( ax) = ap(x) for a ~ o. Let f(x) be a real linear functional f(x) defined on a linear subspace M of L such that f(x) ~ p(x) on H. Then there exists a real linear functional F(x) on L such that F(x) .::: p(x) for all x in L and = f(x) in M. F(x) The inequality F(x) ~ p(x) implies F(-x) ~ p(-x) or -F(x) ~ p(-x). Hence -p(-x) ~ F(x) Theorem 1. 6. 4. ~ p(x). Let L be a locally convex real 1. t. s., D its directed system and M a linear sub-space of L. Let f be a real linear functional defined on iVI and bounded with respect to the pseudo-norms Then there exists a real linear functional x~c(x) on 1 such there. that x~c(x) = f(x) on M and furthermore :0<(x) is bounded with respect to the pseudo-norms (with the sa.rn.e bound). Proof. and M( d) p(x) By the corollary to Theorem 1 • 6. 1 there exists d in D > 0 such that I f(x) I ~ M( d) II x II d for all x in M. Taking = M( d) II x II d' Theorem 1. 6.3 shovrs the existence of x:<(x) and further that -M(d) Since I\ -x II d ~ II -x II d = II x II d' Theorem 1.6. 5 ( 6 ). .:0~(x) ~ M(d) H x ll d. l x~~-(x) I ~ M( d) II x II d for all x in L. If L is a locally convex real l.t.s. and D the associated directed system, then for any x 0 in L, and any d in D, -21- there exists a real linear continuous functional defined on L with the property x'c(x ) = II x 0 p n 0 II d Let d be in D. "roor. f:. 0 and I x-:~- (x) I ~ II x ll d all x in ·L. It can clearly be assumed that II x 0 II d -/= O. Then the elements of the form ax , a real, form a real linear subspace 0 },f of L on vvhich f(x) is defined by f( ax ) 0 = I a I II x II 0 d • I f( ax )1 0 Hence by 1'heorem 1.6.4, there exists an extension x-:~-(x) of f(x) to 1 such that x't(ax ) = a 0 Furthermore = a II x 0 1/ d and II x 0 II d and so x'~- (x 0 ) = II x 0 II d• I x'c(x) I ~ II x II d for all x i n L. This shows that there exists on L infinitely many linear functionals bounded w:i. th respect to the pseudo-norms. Corollary. L. Then :0'(x) Let L'' be the set of continuous linear functionals on = 0 all x-l< in 1 implies x = 8. It is clear that Theorems 1. 6.3 and 1.6.4 may still be used to prove the exi.stence of real continuous linear functionals on L even when L is a locally convex complex 1. t. s. but they no longer apply to complex continuous linear functionals on L. The extension of Theorem 1. 6.4 to the case of locally convex complex 1. t. s. and complex linear functionals is obtained as in the Banach space case ~ 6). This theorem vin ich will be called Theorem 1. 6. 6 is obtained from Theorem 1. 6. 4 by replaci ng real by complex in t he wording of that theorem. Proof of Theorem 1.6.6. linear functional gi ven on M. Let f(x) = f 1 (x) + if 2 (x) be a complex f (x) and f (x) are real linear 1 2 functionals and a simple calculation shows that ) f (x) I ~ M( d) 1 II x II d -22- and l f (x) I ~ M(d) 2 II x ll d on M:. Further, f (x) = -f (ix). 1 2 Theorem 1. 6.4 may be used to obtciin a real continuous linear functional F (x) 1 on L such that F (x) = 1 (x) on M and 1 1 Set x"'c(x) = F (x) - i? (ix). 1 1 l F1 (x) I ~ M(d) II x li d on L. Since x-l 0 (ix) = ix.,'"(x) this is a complex continuous linear functional coinciding with f(x) on E. =re i8 I.f x-:'.-(x) , then ,,_( \ I I x" :x:; Theorem 1.6.S still holds for locally convex complex l.t.s. In constructing the auxiliary functional f(x) the real number a is ~ replaced by a complex variable is used instead of Theorem 1. 6. 4. ranging over K and Theorem 1. 6. 6 'i'his analogue of Theorem 1. 6. 5 will be referred to as Theorem 1.6.7. Further, with these changes the corollary carries over to Theorem 1. 6. 7. Let r:c be the set o f continuous linear functionals defined on a locally convex 1. t. s. 1°< is a linear system. A bilinear functional 3(x,x3<-) is defined on L x L-ic by B(x,r<-) = x"'{-(x) having the properties (3) (i) B(x,x'c) = 0 for all x in L implies x~~- = 9-l<- (ii) 3(x,x-:'") = 0 for 2.ll x" in 1-:<- implies x = e. -~.:._ ' and ~,_ The forner is merely t h e definition of the zero element in L" vmile the latter is a consequence of the corollary to 'l'heorem 1. 6. 7 in the complex case and of the corollary to Theorem 1. 6.5 in the real case. In either event for each x-l'.- in T-)~.LJ is a triangular pseudo' nonn on L and by property (ii) defines a locally convex topology on L. -23- I x'~- (x) l is a triangular pseudo-norm Similarly, for each x in L, on L°'' defining a locally convex topolo gy on L. Def inition 1. 6. 2. The locally convex to pology defined on L by the triangular pseudo-norms topolo gy on L. I x.,<(x) I , £-:.- in 1-:~-, is called the weak The given t apology on L is cal led the strong topolo gy. Definition 1. 6.3. Th e locally convex topology defined on 1-:< by the triangular pseudo-norms I x-:<(x) I , x in L, is called t he weak-i< top olo gy on L.,<. Theorem 1.6. 8 (1 7). A set S C L, a locally convex l.t.s., is bounded in t he strong topology if and only if it is bounded in the weak topology. Proof. The necessity is clear. As for t he sufficiency of the condi tion, S vreakly bounded means sup I x7<(x) I oo, for each x-:~- in L\ x in S Let d be any element of D. Consider all x'< satisfying I x-~-(x) I ~ M( d) II x II d < < These certainly satisfy sup I x-:<(x) I oo. Let x in S • N is a linear sub-sp ace of L. Then L can be for all x in L. written as L in M, JI y = N + M where M is a linear sub-sp ace of L, and for y II d = 0 implies y 8. Hence II • fl d is a norm on M and by the theoran of uniform boundedness for i3 anach spaces ( 2), II x II d ( "" f or all x in s. Let Land L I 1 be two locally convex l.t.s. with associated directed systems D and D respectively. J (L,L 1 ) , the set of all continuous linear functions on L into L I is clearly a linear system. P'o-' ( L,L I) may :; be topologized in various ways, three Qf which will be dealt with here. - 24- If E and F are any two directed systems then the set G of all = (e,f), e in E, f in Fis a directed system if g > g pairs g gI = ( e I , f I ) is defined to mean that e ) following choices for E and E 1 >f • I e t and f 1 , where The lead to directed systems G by means of which the desired topologies may be constructed. ( i) of L. Let E be D' and F the family [ B } F is a directed system if B 1 > B 2 is defined to mean B1 C B 2· (ii) Let E be D' and F the family elements of L. of all bounded sets B { p~ of f inite sets of F is again a directed system i f P1 > 1? 2 means ~ 1 c p 2· (iii) { ~ -l< } Let E be t h e family of all finite sets elements of ( L' of t, where (L' ).;:- is the linear system of continuous linear functionals defined on the locally convex l.t.s. 1 1 • cp 1 -~:- > p2-l< means p 1 -~ C p2-l<. directed set if E is a F is ta.ken as in (ii). Theorem 1.6.9 (10, 15). If the following pseudo-norms correspond- ing respectively to the choices (i), (ii), (iii) as directed systems are defined on (i) the o-J j I (L,L ) '7b pseudo-norm: II T II (d' B) = ' (ii) the J s pseudo-norm: <f? )= JI T ll (d' ' and (iii) the -J ·rv pseudo-norm: s:ip II T(x) II d' T in J (L,L'), x in B s1:1p II T(x) ll d 1 ,Tin ) (L,L') x in 4i II TII ( .+. -i< ~ )= s:ip 1: ' 'i:' \ y1~ fT(x)]j ,Tin J (L,L'), x in ~ in ~ -k y-:~ -25- .7' (1,11) then in each case a locally convex topology is defined on e7/ referred to respectively as t.he J' (1,L'). If ~ b' J <r:' J s and er:- . :J W topologies on ~ ~- are each taken to contain one point and only, equivalent pseudo-norms are obtained. -- Theorem 1. 6.10 (1~. 'flith the same notation as above if 1 is an -:.f -space and 1' is sequentially complete then the linear system J (L,1 1 ) is sequentially complete for the Theorem 1. 6.11 (10). "J b topology. 'Ni th t he s ame notation as above if L is an ':J -space, 1 1 any locally convex 1. t. s. and H a subset of J (1,L'), then the following statements are equivalent: (i) H is bounded in the "J b topolo gr, (ii) H is bounded in the J s topology, and (iii) H is bounded in the § 2. 1. 7. - J W topolo gy. Locally Convex Topological Linear Rings ivlultilinear ?unctions. 2 A function of n variables f(~,x , ••• ,xn) on 1 1 ,1 , ••• ,1n to 1, where 1 , ••• ,1n are locally 2 1 convex complex (real) l .t.s. will be said to be complex (real) multilinear if: (i) f(~,x , ••• ,xn) is additive in each xi' 2 (ii) f(~ ,x , ... ,xn) is complex (real) homogeneous of de gree 2 one in each variable. Definition 1. 7. 1. If D, D , D , ••• ,Dn are associated directed 1 2 -26- systems for the locally convex l.t.s. 1, 1 , 1 , ••• ,Ln respectively, 1 2 then the multi linear function f(~ ,x , ••• ,xn) is said to be 2 continuousat(y1 ,y , ... ,yn)ifgivendinDand 2 (i) > 0 E: there exists d1'd , ••• ,dn belonging to D1'D 2 , ... ,Dn respectively and 2 ~ 1 > o, ~ 2 > o, ... , d n > 0 such that II xi-yi /l d. < ~ i' l = 1, 2, ••• ,n implies j_ II f(:x:.i,x2,•••,xn) - f(y1,Y2,•••,Yn) /I d < E ' (ii) bounded vvi th respect to the pseudo-norms i f to any d in D there corresponds di ,d 2, ••• ,dn in D1'D 2, ... ,D respectively and a 11 real nun1ber M( d,d.i ,d , ... ,dn) 2 > 0 such that all x. in L .• l l The notation of Definition 1. 7.1 ·will be used in the following. Theorem 1 • 7. 1. A multilinear function is continuous everyvfhere i f and only if it is bounded with respect to the pseudo-norms. Pro of: The condition is necessary. Indeed the continuity of f(xp ••• ,xn) at (e, ... ,e) implies, for given d i n D, the existence of o, 1 ~ d. in D. and rnmibers b. ) l l l ll f(~ , ••• ,xn) II d ~ 1. Let t 0 i ~ n, such that for II x.l II di be a real number satisfying 0 For every point (x,,,x , ••• ,x) with 2 I n l Hence it is true that k1 + k2 + ••• +k n II f ( x., ' ••• 'x ) II I Il b., l < I t 0 1 < 1. II x.l 11 d . -/:. o, i =11, ••• ,n, there (. exist n integers lc,k , ••• ,k such that b. I t I ( ll t J. x. II d --, 2 n i o o i i It 0 I ~ d ~ 1 ' and using ~ b .• i -271 1 II x.l II d . , l 11 f(x., ••• ,x) II n I d 1 1 ~ ... 1 1 • This result is still correct for those the prop erty that points (4 ,x , ••• ,xn) having 2 II x.l 11 d . = 0 for some x.l since t his implies II f(~ ,x 2 , ••• ,xn) II d = O. l Otherwise there exists some real number a such that II f(~, ••• ,axi' ••• ,xn) li d= I a I while II axi II d. = I a I II xi II d. = 0 l and I/ f(~, ••• ,xh) ll d > 1, II ~ II , ~ k -/: i. l bk' '\ The condition is sufficient. For n L f(~, ••• ,xn) -f(y1' ••• ,yn) = f(y , ••• ,y. ,x.- y., x . , ••• ,x ), 1 i -1 i i i+1 n i=1 and given d in D, i f. > o, it is true t hat II x.y. II d < ~ ., l l . l l = 1,2 , ••• ,n, implies II f(y1 , ••• ,y.i -1 , x.y., x. , ••• ,x) fl d i i i+1 n ~ M(d,rl. -1 , ••• ,dn) II y1 11 ~ •••• ll Y·_ i 1/I, a. II x.- y. 11 d jJ x.+1ll d ••• llx,n IJ d ~ M (d,~, ••• ,dn) ~ i ( ll yk ll n k -/: i i -1 k + i ~ k). i i . i . i+1 n -28- ~ 1 = ~ 2 - • ". = Setting and C = sup II yk II d = n k k ' it foilONS that II f(x.1'x 2, ••• ,xn) - f(y1'y 2, ••• ,yn) li d ~ n• M(ct,ct1 , ••• ,dn) ~ (C+ h )n-1 ~ when Hence f (~, x , ••• ,xn) is continuous 2 is sufficiently small. at (y ,y , ••• ,y ) and so everywhere. 1 2. n If f(Xi ,x , ••• ,xn) is a multilinear function on the locally convex 2 l.t.s. 1 to itself Definition 1. 7.1 (ii) implies the following statement more suitaole in that situation. there exists d I in D such that n,1 i Since D = D = D = ••• = Dn 1 2 > d. and II x II d l 1 )' II x II d., l = 1,2, ••• ,n, for all x in 1. Definition 1. 7. 2. If f (~ ,x , ••• ,xn) is a multilinear function 2 on the locally convex l.t.s. 1 to itself and Dis the associatsd directed system then f is bounded with respect to the pseudo-norms if' to each din D corres ponds d 1 > O in D and a real nu."!lber :ti:I (d,d 1 ) such that M(d,d 1 ) II x. II I 1. 8. d1 II x? I/ 1 ••• 0 x I/ - d n d1 • ~xtended Locally Convex Topolo gical Linear .i:tings. Definition 1. 8. 1. Ol is a complex linear ring i f is a comp lex linear syste:n and if, in addition, there is defined a multiplication of elements of ril1 • "· 2· such that : Every pair of elements x,y in x,y in l\.f Ot (fl has a unique product {J[ • Hultip lication is associ ative, i.e., x(yz) = (xy)z, x,y,z in (Jl • -29- u3 • = xe = e for The re exi sts a unit element e such that ex eac h x in {!( ~ x) ( "( y) = ( ~ 7 )xy with ~ 7 s. ( D. Multiplica.tion and e.ddi tion a re r elat ed by , in K, x,y in Ol + Z) = Y."\T + XZ X(\CJV .. ~e) ( y + z ) x = yx + zx ()L is a real linear ring if Ot is a real linear sy st em and the postulate S is rep laced by S '. (ax) (by) = (ab)xy, Definition 1. 8.2. a,b in R. , Ol topological linear ring if x,y in O(; • is called. a locally convex complex (real) (![ is a loc al ly convex complex (real) l.t.s. and i f multiplication is a cont inuous function of both variables un der the assi gned to po logy. By Theorem 1. 7.1 a ppUed to the continuous multilinear function to xy on (Jl , where (![ is a locally convex comple x (real) topological line ar ring and D i s the directed system for corresponding t o each d in D there is a d M( d,d') ) I Ol , in D and a real number 0 such that l/ xy ll d ~ 1 N: (d,d ) Hx ll d' // y // d,. In t h e case oi' the loc a lly c onvex oo mplex (real) topological linear ring Oi it is desirable to have t h e stronge r property that for d in D and all x,y in Ol II xy II d ~ Replacing M( d) II x II d II y II d • Hx II d by the equivalent pseudo-norm M( d) // x /( d a p seudo- -30- norm satisfying t h e inequality for each cl in D, is obtained (Theorem 1.3. 2). Theorem 1.8. 2. /I "X'J II d ~ II x /I d II y II d' for The inequality eac h d in D, is equivalent to the existence in the locally convex complex (real) topolo gical linear ring of an equivalent set of c ornrex, circular (symmetric) and balanced nei ghbourhoods { C i of the ori gin, such that for each C I x,y belong to C } CC C c, when CC = [ xy Proof. U -:cy C .:= Let D be the directed system for II d ~ II x II d II y II ct• and suppose Then suitable nei ghbourhoods C are given by t x I II x II d < ;;- < 1 } < ~ < on al 1 values 0 (}[ • as d ranges ov er D and S' takes 1• Conversely, if such a neighbourhood. system exists, then defining I/ x II C = inf l h for each x in II xy II c Ol ~ JI x II c and C in II y II c· numbers a and b so t hat II y I/ C f:- < I) y II C + b respectively. ) { C 0 x is in hC J , it follows that ~ For g iven /I x II C ~ ~ J a > o, there exist positive < II x II C + S , while x and y are in aC and bC Then -:cy is in aCbC = abCC C abCC C ( l/ x ll c +b ( II y II C + S ) C and consequently II xy II c ~ II x 11 c But is arbitrary and so // y II c + ~ ( /I x II c + II y II c) + ~ 2 • /I xy II C ~ II x II C II y II C • -31- The question whether or not every locally convex topolo gical linear rin g contains an equivalent system { C J of nei ghbourhoods (A) satisfying CC C C is ansvrered in the negative 'oy the space 1 oi' Example 1 • h. 1 • Definition 1. 8.3. M_ linear ring alent system A locally convex complex (real) topolo gical will be called extended if there exists an equiv[ C ~ of convex, circular (symmetric) and balanced neip;hbourhoods of the ori gin such that CC C C. This addi tional condition insures, for example, that xinverse of x in fX ) exists for some x in ()l 1 (the and is a continuous function of x on the set of elements having inverses. It also supplies examples to illustrate the theory develop ed i n Chapter III. 1.9. Exa.'11ples of Extended Locally Convex Topological Linear .ri.ings. Example 1. 9.1. Let D be a directed system and let Ol d' d in D, be a f anily of complete normed linear rings which as Banach spaces satisfy the conditions (i) and (ii) of Exanple 1.4.1. (!{, = Then n ot d is an extended sequentially complete locally convex D topolo gical linear ring. With the same notation as in Example 1.L . 1, if x,y are in Gt , II xy II d ~ II x !I d II y I/ d for every d in D. As a particular instance of this cnnstruction D may be taken as = <. p ~ oo • Let a the set of real nun1bers 1 ...... satisfy { a.l ~ 1 L/ a.J_ /P)p < 00' that is { ai } is in lP. A product II a II p i of elements a,b in lp is defined by ab f a. ) { b.J_ ai bi ) • Then =( = l 1 =[ -321 1 ( L I a. I P) P(LI b. I p) P = II a II p II b II p • are now t.1-ie normed rings lp , and 1 = ti lp II ab II p = (Li ja.l b.l I p)p The normed rings {)[ 1 < ..._ l l p 1 ~ p ~ o:> is an extended sequentially complete locally convex topolo gical linear ring. Ano ther example of the same construction i s the following. Let G be a compact topological group and f ,g real-valued functions in G. The convolution of f with g is def ined by (f-;cg)x = Sf( g ) g (y-1x)dy, where x and y are in G anc.l dy denotes the element of Haar measure. The LP s pa ces on G for 1 ~ p ~ "" constitute complete normed linear ring'S under t he usual definitions of addition and scalar multipli cation and with conv olution as multiplication of elements and hence qualify as a farn.i.l>r " N as described above. VL p Example 1. 9.2. Let D be a directed systan and let Ol d, din D, be a family of complete normed linear rings which as linear spaces s atisfy the conditions (i) and (ii) of Example 1.4.1 as modified by the not ation of Examp le 1. 4. 2. Let (/[ be the cartesian product 7T = f xd J , din D. Addi ti on and scalar multiplication of elements in ()i Ol d' d in D, that is the set of all elements x as in Exa:m.ple 1.4.2. given by xy = Multiplication of x = f xd} [ Yd ~ = [ xdyd i • are defined f xd ~ , y = Hence, since II xy ll d = II xdyd II ~ II xd II II yd II = l/ x ll d l\ y /l d. [ yd } is /I x II d = I/ xd {/ , Thus Ol is an extended sequentially complete locally convex topological linear ring . :Sxai11ple 1. 9. 3 (t). Let 4 be any locally compact Hausdorff -33- topolo gical. space and { C The family Let '7r [C j J the family of compact sets of ~ • > c2 means c1 c c2. forms a directed set i f c1 denote the family of continuous nu.,'Tlerically valued functions A . defined on ~ is a linear ring under the usual definitions of adcii.tion, scalar multi plication and multiplication of elements of cg, For each C in [ C j , a pseudo-norm on II f II !" = sup x in C v I f(x) I ' Ji is defined by f in 'J. , • giving rise to a sequentially comple te locally convex topolo gical linear ring . II f g II C = sup / f g (x) x in C I = sup / f(x)•g(x) I ~ sup I f(x) I sup / g (x) I x in C x in C x in C = 11 f 11 c II g II c. ~ In particuJar, let EY..ample 1 • 9. 4. is used. lJ be t he space G of Example 1. 4.3. In this examp le the space o8 of Exa~p le 1.3.1 is the non-normable sequentially complete Jr -space of infinitely d:if ferentiable functions defined on t he interval 0 ~ t of t he real line. Let addition, scalar multiplication and multiplication be defined as usual. This means that for x,y in f unction def ined by :xy(t) = x (t). y (t). II x I/ k = sup I x(i) (t) l , O ~ i ~ k 0 ~ t ~1 // :;.'Y /l k ~ IJ , xy is the Rec2lling that for x in /) , k an integer it is true t ha t ~ 1 ll x ll k II y // k. For, -34- II xy II k = I xy ( i) ( t) I sup o~ i~ k 0~t~1 and xy(i)(t) = sup Now d~ ~ ( ni ) x(n)(t)y(i-n)(t). x(t) y(t) = dtl L_, n:::O / x(n)(t) f ~ 0 ~ t ~ 1 sup o~ t ~ 1 Hence sup / X'J (i)(t) / 0 ~ t ~ 1 and so = II x JI k i f 0 ~ n ~ k. / x ( m) ( t) / O ~ m ~ k II xy II k = = 2k I/ x I/ k /I y I/ k" ~ ll x ll k II y ll k(1 +f + i(~- ) +... + 1) 1 i sup k O '< l. < ' 0 ~ t ~1 / xy (i) ( t) J < - sup 2- ( 0 ~ . ~ k) l 11 x JI k II y II k Putting /I x II ~ = 2k I/ xH k equival e nt pseudo-norms are ob t ai ned such that II :xy /I k ~II x ii k II y/I k" -35Chapter II Analyti c Functions of a Complex Variable 2.1. Introduction. The purpose of this chapter is to develop comp lex function theory in three (related) situations. In§ 1, the t heory is developed in detail for the case whe n t he function values lie in a sequentially comp lete locally convex complex 1. t . s. The case when t h e function values lie in the linear space of continuous linear functions on one sequentially complete locally convex comp lex 1. t. s. to a second such space is sketched in § 2. Finally, in § 3 some theorems relating to functions of sever2l comp lex variab l es taking their values in a sequentially complete locall;{ convex comp lex 1. t. s. are given. 'i 'he p roof s in Chapter II rest heavily on 'I'heorem 1. 6. 7 and its corollary and. on The orems 1. 6. 8 and 1. 6. 11. § 1. 2. 2. Vector-Valued Functions of a Complex Variable Continuity and Differentiability. In this section the linear system structure of the sequentially complete locally convex comp lex 1. t. s . under consideration vvi 11 be denoted by L. 'iihen referring to this spa ce in its strong ( given) topo lo gy , with respect to which the sp ace is sequentially complete, the symbol L "VI.rill be s used and the corresponding associated directed syste11 will be denoted by D. i'lhen referring to this space in its weak topology, with respect to whi c h the space is not necessarily comp lete, the symbol will be u sed. The symbol iJ 1w will always denote a domain, that is, an open, conn ected set in t he comp le x p lane . -36- Definition 2. 2.1. in 1 then x( vvi th values J ) is cal led a vector f unction of Definition 2. 2. 2. (i) If x( ~ ) is defined on given d in D, and E- > k ) on A vector function x( strongly or 1 s continuous at • /3 ~ = ~ 0 in is lJ , if o, there exists ~ = ~ ( ~ 0 ,d, f such ) - x( k 0 ) ll d < € ' weakly or I,N continuous at f = t in o{) , if given 0 ~ that (ii) f x':- in L':- , 0 I<~ implies ~ > o, there is a I ~ - ~0 I < a II x( t == ~ ( ~ ,i::-, f ) such that 0 I x-l:- [ xc .t ) - xc r 0 ) JI < implies E , and oe!J (iii) at each point of o8 . Definition 2.2.3. A vectcr .function x( k ) on J) strongly or Ls (weakly or Lw) differentiable at S == S is 0 i f there S ) in 1 such that the di ffe rence quotient ; { x( 1 + ~ ) - x( t tends strongly (weakly) to x' ( t when 1 O. x' ( S ) is called the strong or Ls (weak or 1w) derivative of x( ~ ) at } x( S ) is Ls(Lv'V ) differentiable in is an element x' ( 0 0 0 ) } 0 ) 4 0 • if it is 1 (1,.r) differentiable at each point of ~ S IV Def inition 2. 2.4. A vector function x( • l" ) defined on JJ is to polo gicaJ_ly bounded i f its values lie in a bounded set. The notions of we2.kly &rd strongly topologically bounded need not be distinguished for vector function s since the notions of weakly bounded and strongly bounded sets in 1 coincide. -37- Theorem 2. 2.1. If the vector function x( ~ ) is 1 s continuous in a bounded closed set l3 of the complex plane, then it is uniformly 1 continuous there, i.e., having chosen s ~ always possible to assign a rnm1ber }. I ~ th-01. t for any tvro points 11 Proof. about eve0J point II sup 11 , s' in C( } ) of x( 11 > ~ ( d, E- ) in such a manner, 0, d in D it is of B for ·which I 3- ~ ' I< 11 - ) 1 x( - ) of J3 as centre such that the x( } ") - x( k ) at l' • .t circle about l ') /{ d < -2€:- , because of the continuity of D let correspond t he Novr to every . -'-h 'NJ.'-' - 1 ra di us "2 r ( .l~ , d , '='/ ) • By compactness a fin i te number of these circles is sufficient to cover B. radius of the smallest of these circles is If the r = r (d, E- ) ' this number satisfies the conditions of t h e theorem. ?or, if / S lies, s ay, in the circle about f as centre with radius ~ r( .f , ct, (; ) , then since d ~ ; r( t ,d, ~ I~ 11 S < ~ , and i f 1 - s S /I d < (A circle C( 1- ) of radius r( 1 ,d, f ) can be dravm /l x( S it is true t hat 1 }- ~ and 6 it follows t hat ~ I J 1 j._ II ~ lie and j.. centre and with radius r( J ,d, E ). 'l'hro rem 2. 2. 2. ~ ~ within the circle about Hence I ~ 11 / x( ~ If the vector function x( '- I ) - x( ~ as ( ) I d < E l- ) is L s continuous on a bounded, closed, con..ri.ected region B of t he complex p lane, then it is topolo gically bounded there. Proof. Theorem 2. 2.1. Given f > O, d in D, determine = ~ ( d, ~ ) by B can be covered. '::iy a finite number, say n, of circles -38- c1 ,c?, ••• ,C n each of radius£. 3 1 any point in B. Let S 0 be a fixed point in B and Join these points to the centres of the respective circles in which they lie. These centres can then be joined by means of a polygonal line defined by r(r ~ n) points ~ 1 , ~ 2 , ••• , ~ r' which are themselves centres of circles Ci, the distance between cons ecutive points being at most JS , l r being the centres of t he circles in which and lie respectively. t 0 and ~ 1 and J Hence, II f ( t ) II d - II f( ~ 0) II d I ~ II f( 1- ) - f( ~ 0) II d = II f( l 1 )- r( ~ 0 ) + r( k 2 ) - f( I 1 ) + ••• + f( J- ) -r( { r) ti d ~ II re l 1 )-f( t :S' (r + 1) E- Thus, 0 ) // d + II re 1- 2 ) -f( l 1 ) II d+ ••• + /I f( 1- )-re .t r) II d ~ (n + 1 ) E • II f( ~ ) II d S II f( l 0 ) lJ d + (n + 1) E- nu.rnb er and } is arbitrary in B. , -where n is a fixed This i mplies t hat for given d E D \ sup II x( JI d < co or x( is top ologically bounded. E B The pro ofs of Theorems 2. 2.1 and 2. 2. 2 use only the fact t hat t he 1 topo logy is local ly convex. k) s S ) Hence t he corresponding theorems for the ~V topolo gy (which is also locally convex) must be valid. 2.J. Complex Integration. In order to obtain a suitable definition of a contour integral t he notion of a Stieltj es integral must be extended to vector functions of a real variable. Only that case in whi ch the integrand is a vector-valued and t he i nt egrator a complex-valued function of a real variable is cons idere d. -39Let x( a) be a topolo gically bounded vector funct i on of the real variable a, and let f(a) be a bounded complex-valued function of the real variable a, both defined for [ a ,b ] 0 0 = { a la a2 < [ a0 ,b 0 ] • define a partition of so that a. sums x n c. ::=: l -1 l ~ 0 ~ a ~ b } 0 • • • • a n = b o, Choosing intermediate points c. l = 1, 2, ••• ,n, the so-called approxi.i11ating a., i l = :f x(c.) [ f(a.) -f(a. ) ] . 1 l l l -1 may be formed. lly the norm i= of a partition is meant the largest of the nmnbers a. - a. l -1 l i • , = 1,2, ••• ,n. If the limit Definition 2.3.1. n lim n-+ co { -7 x n lim n-+ co ( -+ 0 0 x(c.) L: i=1 l exists (call it x) in the L (1vv) sense independently of the manner of 8 partitioning an d of the choice of t he numbers ci' ai_ 1 ~ ci ~ ai' then the limit is called the Ls (11v) Stieltjes integral of x(a) ·wi th respe ct to f(a) fro m a x = ob S a to b o and is denoted by 0 x(a) df(a). 0 In order to show t hat t his definition is satisfactory the follOYfing theorem is necessary: Theorem 2.3.1. Let x(a) b e an 1 8 continuous vector function on [ a 0 ,b 0 ) , and let f(a) be a complex-valued funct i on of bounded va riation on the s ame interval, t hen t he approximating sums x in the L s topolo t;1J and hef'...ce x exists. n converge -40- Sine e L is not necessarily sequentially complete in t h e 1\v topology t he exis tence of the L,,r Stieltj es integral is not ensured .I wh en x(a) i s mereJy LvV continuous. Hovvever, it so happens that in the case at hand the t wo notions of Stieltj es integral coincide. The proof of Theorem 2.J.1 is based on the following lennnas in which it is assumed that the hypotheses of that theor em are valid. Definition 2.J.2. If I is any sub-interval of [ a , b ) 0 and 0 d is any member of D then by t he d-oscillation of x( a) on I is meant o- (d) =sup II II x(a 11 ) - x(a 1 ) li d' a , a I • .,. in J.. = V(a' , b') d enote the total variation of the function f(a ) on t he closed sub-interval [ a' ,b' J of [a ,b ~ 0 0 Lemma 2.J.1. and t he d-oscilla ti on of x( a) on t ba. t sub-interval for given CJ ( d) d in D. Let V s1 and Th en for two approximating smns [a', b' J fanned on the sub-interval is true that Proof. for n = 1 and n = p( ?: 1) it sp - s1 11 d < v o- Cct). 11 Let s 1 = x(c Sp = x(c ) [ f(a ) - f(a 1 SP' which arc 1 0 1 )] ) [ r(b 1 1 ) - f(a )] and + x(c ) [ f(a )- f(a )] + 2 2 1 + xr\ c p)\ ( f ( b ' ) - f ( ap_1 )] be the t wo approximating surrs. Here a , a , ••• ,ap_ denote the 1 1 2 partition points and c ,c , ••• ,c the intermediate point s. 1 0 p hypothesis II x(ci) - x(co) ll d ~ u (d), i = 1, ••• ,p. By Since s1 can be written in t..he form s1 =x(c 0 ) [ r(a1 )-f(a')] +x(c 0 ) [ r(a 2 )-f(a1 )J +••• +x(c 0 ) [f(b )-f(ap_1 )] 1 i t follows that for d in D , -41- II Sp- s1 11 d ~ ~ (d)( l f(a1)-f(a 1 ) I + l f(a2)-f(a1) 1 + ••• + l f(b') -f(ap-1) 1 ). ~ CT ( d)V. Lemma 2.3.2. Let Sn be a fixed a pproximating sum on [ a ,b ] 0 0 and for d in D let the d-oscillations of x(a) on t he n segments of the ,,Id) • v 0 \. partition all be less than sum derived from S n Then if V on IJet S be a new approximating by adding new partition points to the old ones. =V(a 0 ,b 0 ) denotes the total variation of the function f(a) [ a ,b ] o o it follows that ford in D II Sn - s ll ur:i ~ V er o (d), no matter how the intermediate points defining S are chosen. L~~ma Proof. 2.3.1 holds for each of t hen segm ents of the partition defining s, so tha t for din D +v n 0- ( d) o = v (} (d) o if v , V , ••• , Vn denote t h e variations of the function f( a) on these 1 2 segrnents. Lemma 2.3.3. E > O, d in D, there exists a S = ~ (d, f Given ) such t hat i f s and s are any t wo approximating sums defined by means 1 2 of partitions of [ a ,b 0 0 J of norm less than S , then Ii s1 - s 2 II d < ; • Choose Proof. = ~ ( E ,d) so that for any two points a ' and a 11 of [ a ,b 0 0 11 1 Hx(a ) -x(a ) li d < 4v(: J for which l a" - a' ) < S" ., This can be done due t o Theorem 2. 2. 1 on uniform conti nuity. approximating sums mating sum s 3 b ) o' o Ni th s1 a nd s 2 as in t he hypothesis, a third approxi- is obtained from them by taking as t h e partition points - 42- defining it those of the f irst two partiti ons combi ned. This third can evidently be consici.ered as derived from them by means of adding Hence, it follows from Lemma 2.3 . 2 that for din D partition points. V• = 4V s II d < V• i... = e 3 4V 4 and similarly II s 2 4 Consequently, f 2. Lemma 2.3.4. Let an approximating sum S n be formed for each If the norms of the respectiv e part itions integral value of n. decrease to zero with increas ing n, then l im S n - > oo Proof. Given ~ > O, d in D, determine E- according to Lem."'!l.a 2.3.3. exists. n = ~ (d, E- ) Take N = N( d, E- ) so large that the norms of the partitions corresponding to the S n with n ~ N are all less than Lemma 2.3.3 is then applicable to all these s ' and n ll s-s l/ m n d < ~<E- for m > n 2 sequent ially complete lim Il-+ 00 less than ~ n m-+ oo It remains to show that i f € > o, S II < ~ holds for every m-+ oo m d for whi ch the norm of the defi ning partition is II x n - lim For in the proof of Lemma 2.3.4, l etting m-+ co it follows t hat II lim m-+ oo II xn - lim sm II Since L is strongly = s ( d, E ) is determined according to ~ Lemma 2.3.3 t hen the relation approximating sum x H. sn exists. Proof of Theorem 2. 3. 1. d in D are g iven and ~ d sm sn II < d = II (S n - lim m _.,. oo ~ I) f 2 for n > N. Hence sm) - (sn - x n ) II d, sn - l i m sm Jl d+ ll s n -xn ll d < m-+ oo t -43Thus lim S n-+ oo x and the ex:L s tence of the Sti eltj es is actually n i ntegral has been proved comp letely. If ~ (a) and x (a) are strong ly continuous vector Theorem 2.J. 2. 2 functions and i f f(a) is a munerical-valued function of bounded vari.ation in a b f a ~ 0 ~ a b 0 , then b 0 [ Xi ( a) + x 2 ( a) ) df ( a) = 0 Theorem 2.J.J. J ~ a 0 ( a) df ( a) + j a 0 b 0 x 2 (a) df (a). 0 If x( a) is a strongly continuous vector function and if f (a) is a numerical-valued function of bounded variation in ~ a \] J a ~ b , a b 0 then for any d in D, b 0 0 x(a)df(a) II SJ d 0 a 0 11 x(a) II d dV(a) ~ 0 sup a ~ a ~ b 0 II f(a) II clV(a ,b 0 0 0 [ a0 ,a ] [ ao' bo ] $ It is clear that V(a) is again a numerical-valued Proof. function of bounded variation in a D, 0 :'S a ~ b 0 and that for any d in 11 x( a) II d is a continuous numerical-valued function in a ~ a ~ b 0 Hence j a 0 II x(a) II d dV(a) makes sense and is an ordinary Stieltjes 0 integral. a a. ), 0 wher e V(a) is the total variation of f( a) on the int erval arid V( a , b ) i s the total variation of f(a) on 0 l -1 ~ < ••• < a n = b o be a partition of [ ao , b ol and 2 c . ~ a. , i l l = 1 , 2, ••• ,n be intermediate points. int erval of th e partition is further sub divided say n t he points b~i), b~i) , ••• ,b~~ ). l [ a . 1 ,a. ] il L x(ci ) [ f(ai)-f(ai_1 )] i =1 Each is an by b • 0 -44- approximating sum to the fi rst integral an c:i. t ne following inequalities are then vali d for ariy d in D. I f, x( c) [ f( ai )-f( ai-1 ) ] II d ~ > f x( Ci) II d [Ir(ai )-f ( ai-1 )u ~ t• 1 II x(c.) II d [ /r(a. )-f(bk(i)) / + ••• + lf(b (i))-f(b (i)) / + /r(b~))-f(a._ )1] 1 1 2 l l= l . I l f: sup // x(a) I/ V(a ,b ). d 0 0 ao,< a,<b o b Hence the inequalities follow. a b as f II a J 0 0 I/ x(a) I/ d dV(a) is s ometimes written 0 x(a) I d j df(a) / • 0 fheorem 2.J.4. Let T be an L s continuous linear t ransformation I on L to L , a sequenU.ally complete locally convex comp lex 1. t. s. r 'rhen under t he hyr;iothesis of theorem 2.3. 1 b T [ a Proof : T []° a lim J= a 0 T [ x(a) l df(a). 0 From t he continuity and line a rity of T it follows that x(a)df(a) ] x(c.) frcai) - r(~~ J ] = T [ lim L i=1 Il -> 0 n-+ cn ( -+ 0 x(a)df(a) 0 11 n = J b C? l l 1t -+ 0 b Li=1 'r ( x( ci) J [ f ( ai) -f( ai-1 )J = JoT [ x( a) ] df ( a) • a 0 -45- Corolla:r;J. r Let be a rectifiable curve oi' length 'f., in the = ~ (a), 0 ~ a comp lex plane g ive n by continuous and oi' joundecl variation in continuous function on 1 r ~ [ 0, a 0 ~ a , where 0 ] • If x( (a) is r ) is any Ls to 1 then the (contour) integral h ~ ) cl ~ II Jr r ) d J-1)d ~ sup llx(S )lid l, for any ct in D. If T is any 1 s r x( x( = a x [ ~ (a) ] d ~ (a) ey,j_ sts. continuous linear transformation on 1 to 1 T [ 2.4 Ir x( ~ )d ~ ] = j 1 ?urther (as in Theorem 2.3.J~) then T [ x( t )] d t • rt e~ular Functions and Cauchy's Inte gral Theorem. In classical function theory a function is regular in a domain i t is single-ve_lued, continuous an d differentiable in /J ~ if The first oi' these :'"lotions carries over without ambi ~·•1ity bc.."t the second and t hird notions have different meanings for vecto r functions depending on which topolog-<J is '-.lSed for L. here a re the Ls and the 1\n r topologies. 'fhe only t apologies considered However, while the notions of continuity and differentiability do have different meanings in the different topolo gies a unique definition of re g 1lari ty is obtained. 1 Definition 2.4. 1. domain ,,!) A vector function x( J- ) is regular in the i f x'~ [ x( k ) J is re gular in the classical sense for every choice of x->< in L-l< I'his is the weakest of several poss i b l e definitions. x( l ) mi ght have been defined to be analytic in strongly dif ferentiab le there. 1J For example, if it i s However tMs can be proved from the -46weaker condition using Theorem 1. 6. 8. Lenuna 2.L. 1 ( 7). 'I'he following Lernma is ne eded. If f( } ) is a comp lex-valued f unction regular in the domai n and .!J 0 is any domain whi ch i s bounded a.YJ.d strictly interior to /) , th en there is a finite po sitive number M(f; .3 in [) I 1 0 ) such that for every choi ce of t + ' o( ' 0 1 { cl..-/3 :: [ f ( 1- + o( ) - f( t Theorem 2.4. 1. domain cf} )] M(f; r/J ). 0 Let the vector function x( ~ ) be regular in the , then x( S ) is L s continuous and L uniformly wi t h r espect to in any domain bounded and strictly interior to .!) Proof. s .() which is 0 • x':- [ x( t )] Apply Lemma 2.4.1 to the function xi:- is an arbitrary element of L*. differentiable in j) , where The lemma t hen asserts I "~f { }<x* [ xP +")] -x* [ x<k l] l -}(x* [ xU+fl ] -:0' [ x( s )Jj ~ I x* [ ol ~ f (x(J + :)-x(.f ) for every c hoice of s- ' J + o( x( S +f;-x( I ) )] ,; J + (3 j) ' in ~ 1 II c< -13 or, 0 and £ ::- in 1-i:-. ' J + .( s +~ ' there exists a finite number M(x,d; jJ ) such that 0 ( x( S + o< ) -x( S ) x( J + (3) -x( 1- o( /3 ~ o) 0 By Theorem 1 • 6. 8, for every d in D and every in M(x'' ,x; M(x,d; ,,!) ) 0 - 47- x( f + /3 ) - x( J )/ x( } + o<~ - x( S ) (3 t I)( n } In particular, let be a sequence of complex numbers such = that lim o< O. For given E > o, d in D, there exists n n-+ °' 3- + r:{ N = N( d, E ) such that for m > n ~ N, ~ + o( m' «'.5 0 and in II x(} o< n / < t /M(x,d; lJ o ). /ti m - + o<. ) - x( } ) m m o( J M(x, d;obl n o )< E • d sequence in L. )J ) x(} + <X ~)n- x( ~ ( "" and the elements Since 1 is complete in the 1 strongly to an element x' ( l Moreover, sequence are Hence, ~ ) o( - o< m n ~ { /3 n ~ For let jS n = O. numbers such that li.m s topoloc;y they converge ) in 1 for any being fixed, x' ( S { o( n ) • form a strongly convergent in o6l 0 • ) is independent of the be any other sequence of complex As before the elements n -+ m [ xU + ft ~ n- x(J" ) ] converge to an element x' ( } ) in 1 for any in <3 0 • For the then and any d in D, swne ll :x' ( t form a convergent sequence in 1 and hence ) - x' c t ) 11 ~ d x( ~ + ~ n) - x( k) II :xt s)- x( J + o( In x( } + o<. ) - x( J- m o< ) - x' ( } ) + m • d P n) - x( ~ ) II + In d -x ( ~ ) ) m o( m x( J- + d -48Novv given n N1 ( d, E ;?. o, din D, there exists N1 (d, 0 ) such that x( f + ;3 ) - x( } ) II > <:: !I x' ( s- ) - implies ) t: II d < 3 • ; n Similarly, there exists II x( l + f3 n ) - x( N (d, E 2 s) such that n,m x( S + « fn m~ : x( l- ) II ~ / B - cl_ / M( x, d; ~ ) Fn E (----6M(x,d; cX'.'.3 ) Finally, there exists N ( d, € 3 II S + o< ) - x( S- ) ~ > ) o JI 0 .>, N (ct, € implies ) 3 -3 . < -x' (l- ) 6 )M(x,d; c<3 0 )= J 6M(x, d; ,JJ- ) 0 such that m m If N( d, E ) 6 + 0 x( m d <:: d = max( N1 ( d, E ), N2(d, E ) , N3 (d, f )), then for m,n ~ N(d, 0 ) II x' c ~ ) - x' c.r )II d < E ' that is' x' ( } ) = x ! ( f ). J It remains now to show that for fixed exists in t he 1 s topology and is just x' ( J ). , l im ex'. -> 0 Givari x( f + o( )-x( } ) o( E > o, d in D II xU + . ;x u ) - x' ( r )II d ~ 11 x( } + <:-x( )- ) - xcf +<>(: nl-x( ; x( t+ o( n ) ~n -x( J- ) - x' ( r ) II d < / oe - o( n / M(x,d; E rb o ) +3 :S ( I o< I + I o<: n I )M(x, d; .1) o ) + -E- 3 f 6 f < -+-+-= 3 3 3 c/ ' t + . -49- for Io{ I < in c<5 (: n > N. Thus x( t ) has a strong derivative J M(x,d; .() 0) and so is L continuous in ./} • ' s In the expression, f -+ o. By the continuity of the pseudo-norm it follows that It x( s +t< )D( -x( s ) - x' ( r ) II Io< I M(x,d; }) ) , let 0 d i.e., the difference quotient approaches the derivative uniformly with respect to ~ in o(_) This in turn implies strong uniform con- 0. tinuity a.rid weak differentiability and continuity. i'heoren oB domain If x( 1 ) is a re g-ul a r vector function on the 2.h. 2. to 1 then 1r J x( )d ~ =e for every simple closed rectifiable contour r the interior of Proof. and T Take 1 . 1 • = X''" in ~(- O = Jr Jl [x( belongs to 1 /) r' in .U such that • of Theorem 2.J.5 as the space of complex numbers Hence, ~ )}d.t = x < [Jr x( S- )d ~] 7 for every x-:c in L1~-. The corollary to Theorem 1 .6. 7 then implies j r x( .t ) d J = e. If ,17 j r x( S )d l is simply connected it follows from this theorem that is independent of the path and depends only on the -50- initial and final points. The procedure taken in arriving at Theorem 2. 4.2 may be swnmarized quickly as follows. A definit ion of regular vector function was given -vrhich threw t his concept back on t he concept of regular function in classical complex function theory. The definition of contour integral on the ot her hand concerned itself only with 1 and J r x( f )d ?" By means of Theorem 2.J.4 which states that con- was an element of L. tj_nuous linear operators (in particular, continuous linear fun::tion als) commute with inte;;ration these concepts were related in Theorem 2.4. 2. At this stage t hen the follovvlng situation presents itself. Methods exist in 1 for proving theorems about the regular vector functions while on the other hand methods exist whereby the proofs of theorems on regular vector f unctions are thrown back on classical complex func tion 'rhis is illustrated in 'l'heorem 2.4.J where t wo proofs are t heory. given. Let x( f Theorem 2.L.•J. on "'6- r If • ) be a regular vector function defined is a simple closed rectifiable curve in interior of which is contained in x( I if )= is interior to Proof: 1. function theory. ,,[) , the , then x( "t) 1 - f , x( z ) ano ----t - .l d 't means ( <: - .r )- 1x( T ) • Here the proof will be throvm back on classical complex Since x'~- [ x( ~ ) J is regular for each X:' in L" then " J t_ -51x-:~ [ x( t J r-r 1 =-- ) J dr r 2 1f i and by the corollary to Theorem 2.4.J X- 0 [ x( S- 1 = i:< [ - 1- )] 2 1f i x"-"- [ x( .1~ or, r -12 Tr i ) - x( t ) t - J- = o, for every x~0 in 1-:<-. x( T ) d T ] t- ~ 1 x( } ) = - 2 /f i or x( il - 2. l =J .r x( c ) r t-J- d 't 2 Tr i • is a re [;,r ular vector function in ) If • 1 M I containing I J 1 r )- The corollary to Theorem 1. 6. 7 then implies x( o!J x( T ) d t = 8 r r - .l except at is any simp l e c l osed rectifiable curve inside /., in its interior then (as in the cla ssic2.l case) J t- x( z- ) d Y x( 'f ) J- T - )- r r In partic ular l e t d ?: • be the circle -with c entre and radius ;; , then x( ~ _ X,r )>- ) - j ) + x( r ) - x( S t -J dt r -~1 l - .J + J r ) d 1: x( r- ) - x( .f I ------- 1:-J- The first expression on the ri ght hand side is 2 Tf i [ x( for any d in D II J r ~ - 2 7T i { x( k x( l t - r max II x( T ) - x( ~ ) \I d r' ) dl /1 - sI )Jii J s )] . Hence, x( T ) - x( f ) d t =I l - J d r' 2 71( < E d -52- ( > o, d in D, there exists This last result holds since 8iven ~ = ~ (d, E ) such that for I/ x( T:: ) - x( s ) II max r' < - 0d = I -r - J I / S < ' • 2 7/ Thus for any d in D it is true that fl j x( 'l ) dt 1 - .J x( f ) = or, i'heorem 2.4.4. ~ domain - 2 7f i 1 2 7r i Let x( to the s pace L. fiable curve in ,/) t )J IJ x( ?" ) dI t -s = 0 d • J ) be a re gula r vector function on the r Let SL~ple be a closed recti- , the interior of whic h is contained in t hen = -n.--. I 2 'I II ~ [ x( l f ------...x( 7: ) ( ~c _ ~ )n+1 r j dY o!J , n=o,1,2, .•• ' It is clea r that fo r every x':- in L~:- , Proof: i'' {x( l ) J .= x~'[ d~ d dJ xU )], where th e derivative on the ri ght hand si de refers to the strong derivative of x( J ) and t hat on t.'le left to the ordinary derivative of the comp lex-valued function x-;< [ x( t )} . ; { x* [ x( by letting o< 'i.'his follows from t hat fact that ~ + d ) - x* [ xd )] } = x* I; [ x( J- + ~ ) -x( } ~ 0 and recalling that x( S ) has a strong derivative. Further, the strong deri va tive i tself has a seconC. strong derivative for x( f ) being a re gular vector function £;';: [ x( !: )) ha.s, for every - 53- x:r, in L~'", derivatives of all orders, anci. so ~ x( S ) is a regular dS vector function. Similarly the existence of higher stronger derivatives may be shovvn and 2 >• -d - x« ds [ 2 d n x-l(- [ -- dt n x( x( l ~ d ) ] =- - -~- [ d x - dl )J 1, xU ) ] • _ ~<- = i~ [x( c r 27Ti -X for a 11 n and -:i" i n L~(-, n 1 i.e. -2..__n x( k ) n. 2 71i J n~ )i - Al so, ct s d2 x( S ) ).Jl = x~< [rd}2 x( f df Ct - S- ) n+'I [' 1 n. 2 7r i r fr ( f x( T 'L - )] d 'L x( r. ) ( Y - .\ )n+1 ) )n+1 • ) is regul ar in I s - s0 I < r, and if for any d in r to be // x( S ) II d ~ M( d) in this region, then taking D, the circle I k - ~ 0 I =r - E , 6 > 0 i t fo llows that If x( r /I x( S- ) /! d ' ( ' 2 1! (r- f) ~ n. M dJ , all r I I - t 0 In +1 ( r- ~ ) n // xn( ~ ) /I d ::: n: M( d) , for any d in D. max o. Hence, o rn The identity theorEE for re g111.ar vector functions is easily obtained. Theorem 2. 4. S. in If x( S and if x( I n ) and y( J ) = y( l n ) are regul ar vector functions ) , n = 1 , 2, ... , the points { rn~ - 54- having a limit point in of) ' then x( r ) is identically equal to y( 1 ). x·~- [ x( _\ ) ] &"ld x-:c { y( l )] for all x·:c in L:c and x'" [ x( :r n) ] = x~:- [ y( ~ n) J Proof: Since are re gular in it follows from the i dentity throran for regular functions in the classical theory that x-::- [x( r ) ] = x"'" [ y( l" )] Hence x( J ) = y( J ) for all all J I in in lJ and x -~~ .v. in L". /) The converse of Theorem 2 . 4. 2 holds in the follovdng form. Theorem 2.4.6. If x( J ) is a stron&:ly continuous vector function on the simply co :Lriected domain to 1 and i f fr x( s )ct S = e for every simple closed rectifiable contour is re gular in Proof: in lJ , then x( l JJ For e verJ x':- in L~" , 0 = x-::- ( 8 ) = .z.::- {Jr x( J )d S ] df r )] I for s ) is a r egular vector function on • By Morera 1 s Theorem -:J:- [ x( .\I_ each x" in 1..,,- and hence x( 2. 5. r is a regular function of Po-v-ve r Seri es anci Taylor Expansion. In what follows a s i tuat ion similar to the one commented on below will prevail. It will be possible to treat re gular vector functions defined by power series by wo rking only in L but it Y.Iill also be possible to throw most results back on the corresponding results f or complex-valued functions. ) -55- Definition 2.5.1. n ( i) If s n be a sequence of elements of 1.• Let { xn ] 00 xr converges strongly (weakly), then [ x r L r=O r==O is said to converge strongly ( weal<1Y). (ii) If for every d in D, ) . II x n=O is strongly absolutely convergent. n II d converges then L / x"'\ xn ) n=O 00 (iii) If for every x-:< in 1-:;-, eo converges then J x is weakly a.bsolutely convergent. L n=O n n (iv) If Sn( J ) = L xr( J- ), -where the xr( f ), r = 1,2,. •• , r=O are vector functions on L 00 xr( dJ , is strongly (weal<:ly) convergent then J ) is strongly (weakly) convergent. r=O Since 1 is segu entially co;nplete in the strong topolo gy then the 11 strong convergence of the partial sums S n =L r=O x r implies these converge strongly to a limit x which will be called the strong sum of 00 "" L =L x is another way of saying x • To vvri te x n=O n n=O n "" this. 3imilar ly, x ( ~ ) strongly convergent vlill also be co n=O n x ( S ) "V'.ihere x( S- ) is the vector expressed by writing x( k ) = the series 2: L=O n function defined by taking the ~trong sum of in ( v) n in JJ L xn ( I ) for each n=O lJ • "" ~ xn ( l ) converges strongl y ( weakly) uniformly for J n=O in any closed subset of sums S ( t oo to a fu_nction x( S ) if' its pa.rtial ) converge strong ly (weakly) uniformly to x( j rVri t ten in full this means: ) for f -56- t S ) is strongly uniformly conv er gent if given xn( n:::O there is determined for any din D, a number n n (d, ~ ) that for all n -?. t 0 x ( l r sup / j ~ in .3 r=O t )- = 0 > o, n (d, E ) such 0 ~ ) // d < E • xr ( r:::O 00 l x ( t ) is weakly unifomly convergent i f given n:::O n there is determined for acy i:'" in ii~-, a numb er n 0 = n ( x~", 0 E E :> o, such • 00 Theorem 2.5.1. If [ xn is strongly (weakly) absolutely con- n:::() vergent then it is strong ly (weakly) convergent. Theorem 2. 5. 2. Let [ xn } be a sequence of elements in L. 00 If L x n:::O sums S n {x 1 • n is strongly (weakly) convergent then the sequence of partial n =L n=O x r is strongly (weakly) bounded as well as the sequence n Proof: Given f > o, for any d in D there exists an integer N = N( d, E ) such that for m > H j JI sm ii d - II SyJ ll d I '< II s m Hence ll Sm I/ d < M(d) =max( II Sn I d < { xn ) E + ll s1 II d' ... ' II SN ii d ' form ~ SN-1 // d' t + ll d < € • > N. Setting IJ SN II d) it follows that M( d), n = o, 1, 2, ••• , and since x is also strong ly bounded. si.rnilar. - SU n = Sn+1 - Sn the sequence The proof for weak boundedness is - 57- Definition 2.s.2. By a power series on K to 1 is meant an expression of the form 00 ( t - l o )n x n 2. 1i n=O ·where x n is a fixed element of L, where } and where t 0 is a fixed complex number If all but a finite number of the x is complex. n are the zero element then 2. 1 is called a polynomial, that is, n L (1 - J )r xr is a polynomial (of degree n if xn-/= El). 0 r=O The first i mportarit question concerning power series is for what values J of does 2. 1 conver ge strongly (weakly) and fo r what values of does it di v erge stron gly (weakly)? ~o will be taken equal zero in the following , but the results holci in the usual manner if l 0 -I= o. 00 Theorem 2.5.J. L >-n x ~ is any p ower series whi ch does n n=O not merely converge strongly an ywhere or nowhere (i.e. only for } = 0) If 00 t hen a definite p ositive number r exists such that s conver ges strongly (indeed strongly absolutely) for f or I n=O sn n / ~ / <. r x s but not n1 > r s· The numb er r s is called t h e radius of strong con-.re r gence. The proof is based on two lemmas. 00 Lemma 2. ). 1. strongly for ~ j If a given power series = ~~ 0 ( >j 0 ~ I n x n converges n=O -/= O), or even if the sequence -58- l j: J xn of its terms is only strongly bounded there, then L j n xn is strongly absolutely conver gent for every en n=O such that f= l 1 r I 1 I <. 00 ?roof : I: If is strongly conver gent then, by The orem n=O 2.5. ?, the sequence there exists some positive nu.iuber H(d) such that o.11 n. Hence II 1n xn /j d - - 1 vrhere {) = I S I < I l j, 11~ < 0 For d i n D i s strongly bounded. 1. /lson x n li d < M(d), ~1 n ro 'l'hus fo r each d i n D, and such that L II S n xn II d is a convergent series of posit ive 00 . n=O real numbers and the result follows. Lemma 2 • .5. 2. If the given power series co L f n xn is not strongly convergent for for any Proof. J 1 with If the series were strongly convergent for then by Lemma 2. 5. 1 i t would have to converge for the point f 0 contradictin g the hypothesis. The pr oof of 'I'heorem 2 • .5. 3 then .follows i n the usual manner. nest of intervals is constructed defining tne nur.1ber r t hat t heor em. s A mentioned in i'his proof however does not su 1Jp l y any inf ori.nati on as to t he magn..i..tude of this number. l'heorem 2.). h. This is oo tained in Let d belong to D and set -59- 1 1 ) rs = l~.L ' fol ( d) = sup µ (d) din D/ s ~ (d) == lim • s J:' ( s d I n 00 L Then t he pow-er series x n n=O Ir I < r s , but does not converge strongly eve"YIJ such that for any sucht h2.t choose r d in D. and so 1 > r s• /S/ k 1 is a ny complex number for which If Proof'. converges stronGlY absolutely for < s uch that r 1 < r • s Now n ,____ J11 x n II d == / Hence given d in D, lim n 1 / II x II < for n n d njll ~ n xn /I d < ~ 1 ? s ( d) ~ -r1 , all 1 s1 < r- , s (ct) ~ r 1 3 Finally, r1 00 1 for n ~ n ( d) and < IS 11 < rs' 0 r1 L n=O }. n J x 1 n is strongl y absolutely conv ergent. On the other hand, if = 1 ( , \ , that is, - s~p )'< s a 1 ~ / 1~ 1 1 some e in D such that n 11" 1/ consequently xn II e > //I < < sup d in D )l n ( d). Hence there exists r s (e) == lim /11 xn II e ' and --J-1 for an i nfinity of n, or I/ > 1 for an infinity of n. n e certainly cannot converg e then. II f1 s x 00 The series L / ~ n xn I/ e n=O The situation with regard to v..-eak conv er gence is similar. Theoran 2. S.S. L '>n- _) x is any power series -vihich does n n=O not mere ly converge weakly anywhere or nowh ere (i.e. only for = 0) If co then a defi nite -pos itive number r w exists such that converges \'ieaJ::: ly (indeed weakly absolutely) for every L n=O J- snxn I :S- j < r w and - 60- does not converge ·weakly for l 'r I > r • l'he number r w w is called the ·weak radius of convergence. Leimnas 2. S. 1 and 2. 5. 2 hold i:vi th i'he proo.i' follows as before. strong r eplaced by weak . In the proof s of these theorems the pseudo- II • II d' din D, are replaced by t he pseudo-norms nor ms x~'> in 1-;c. J x-:c(•) l, The radius of weal< convergence is obtained in Theo rem 2. 5.6~ Let x -~~­ belong to L::- and set n J Ix~'> (xn ) I ' r w ==x~'"infin 1-:c (/ v/1-.;;.-;;-\ ) = x~'> / ~w(xi:-) == lim 1 }n x converges weakly absolutely for every n n=D Ir I < r w but does not converge (weak ly) for any 1 Then the ?OVrer series such that I~ I > r • w such that Theo rem 2. 5. 7. The numbers r s and r w are equal, that is, the radii of strong and weak convergence are the s am e and equal r say which vri.11 be referred to as the radius of convergence. The notions of strong and Heak convergence for power s eries are equivalent and need not be clistinguished. D ~ .r roor. It is clear that strong convergence i mplies weak but the t n x is weakly convergent for { = n n=D x ~) is weakly bounded and hence, by Theorem then the sequence o n >- n x Thus for ·/ ~ I < I L\ .J is 1.6.8, strongly bounded. o ' n=D n stronr:;ly convergent (Lerruna 2.5. 1 ). converse is also true. [f If 1 00 I .\ Theore:n 2. 5. 3. 'l'he power sGri es t n=O i,_n j x n r epresents, for S 0 - 61- l r I < r, i ts ra. c.ius of conver genc e , a re;_;ula.r vector i\mc tion :x( f ), 71hose derivatives are obtained by cii fi'erentiating the power series term by term an<i t hese derived power series have the s arne radius of convergence as the given series. Proof: --- I~I For < r 00 and each x ~~- f L: n:::O in 1-:<, nx'(- [xn ] is a convergent power series with complex coefficients and hence defines Further a regular function of a complex variable there. N 00 x-:~- (x( 1 )] = x-:<( L 5n x ) = x-:< ( lim Z: J x ] n=O n N n~ 00 n=O n co I: Jn x'"( x J = L Sn x~- {x J for = lim n n n N ~ oo n=1 n=1 Hence x( J- ) is a regular vector function of Exa~ples I ~ I ..::.. J r and each x in HI <'.'. -~~ " in Li"• r. of regular vector functions are easily obtained. For J the space of integral functions (1 9). lor each 00 n n z function and S ~ is a povrnr series on is an integral n, -n n=1 n the complex numbers to Its radius of conver gence can be computed instance let L = n L J II zn II l zn / = m11 and hence //. ( m) = lim ~ = O. Thus moo lz l = m n - s n ~ 00 n r = oo, and j 11 ~ represents a regular vector f unction (vrith values But, = sup l: in J n=1 ) for all (ii) element of n } Let x(z) J . • · ... al f unc _,_. = L~ anz n b e any invegr vJ_on, that is, an 1 n=O This is an integr al function if and only i f I a In--+ 0 n - 62- .L.ver;;r continuous linear functional x~'c in as n-+ ""• form L_ cn a where x( z) n' >. n zn on K to J n n=O n = I_ a n z . Here n t he following results are v a.lid. / x~:- ( z:) 1= lim i /l ~\zn) I = 0 since {1 en I 11 TI-+ co = co len 1n Treating a gain the power series 1 r is of the n=O n=O must be a bounded sequence. ~ -~~- 00 (X) x~:- [ x( z )] J 1 Thus is touncted, as before. Accordin3 to Theorem 2 • .5.8 , within its radius of converg ence, the (X) p .:ywer series >- n L x represents a regular vector function of the n n=O j com;,Jlm: v2.riable J • The converse statement is contained in the follovving theorem. Theorem 2• .5. 9. dorn2.in S- ) be a re gular vector function in the Let x( and let J 0 be an interior point of is one and only o ne power series of the form conv eni:es for a certain nei r;hbourhood of function x( oJ l'hen there (X) J E x ( I - J )n which n o n=O and represents the 0 S ) in that neighbourhood. Uoreover, xn - xn( J o) • The I n. series converges at least in the Jargest circle which contains only points of Proof : ~ about J0 • For each x-:'c in 1-::-, x~:- [ x( l )J is a regular function in J 0 and so c an be expanded in a Taylor s.sries about the point converging (absolutely) at least in -16 (X) x~'c [ x( ~ ) ] = [ n=O -::- [ x (n) ( X 1o) I n • 1 ( ). ~ - ~ ...\ 0) n • - 63- The absolute convergence of this series in ~ co x(n)( } o ) . J )n in ver genc e of ( 2n=O s- n. imp lies the con- _,,e with sum x( J 0 The uniqueness follows from i'heorem 2.4. ). s. The pro of of t.11e fo llowing theorem on uniformly convergent sequences of regula r vector f unctions agai n illustrates the technique of throwing proofs back on the clas sictl case. Theorem 2.5. 1 O. func ti ons on to J Let x ( n s) to L ·which conv er ge strongl y uniformly ·with respeet on a simple closed recti fiable curve say, is also in ,J analytic fun:; ti on x( .f ) in /) )} in be a sequence of re6ular 0 ~ 0 f or every k. Th en o' , the interior of xn ( J ) converges to an and moreover, xnkr' J ) -+ xk( J- ) The conver gence is uniform vri t h respect to in an.y fixed closed domain inte rior to Proof: r1 l ,,() o· x ( J ) -+ x( J ) in the strong topology implies n x ( S ) -+ x( J ) in the weak top olo gy and so lim n The sequence { x''c [ xn ( t )] } x"'"[ xn ( S )) = x·:<-r x( J- )} • n-+ "" being uniformly convergent ~"[x( S ) ] is re gular for every x-:c in L-:'c and so x( i ) is a re gular vector function. Similarly the r est of t he theorem f ollows. The pr inciple of t he maxirrmm can al so be extended to vector valued functions. Theorem 2. 5. 11. bounded domain Let x( S ) be a vector f unction defined on a o!J strongl y continuous in and on its boundary .!J v 18 • t3 , regular in ,,& and If for arry d in J , sup I/ x( } ) // d = M( d) -64- for J on j) u lf3 then either /I x( l" d3 or ) /I d = E( d) for all f I/ x( S ) /I d < M( d) in <!) • Proof. The proof of this theorem needs the result that i f f( a) is a continuous real-valued function of a real variable, f(a) lb 1 and b _a 0 0 0 a ~ k, then f(a) f(a)da 0 Let r ~ /I x( S ) JI d' ~ 0 I )= 1 0 S0 of J , Hence, S being any other point in olJ u di • I t -J r II x( S0 ) /I d ~ 1 or, Then and radius r. x( t ) 2 71 i k, 0 be a circle which lies entirely in centre at x( d < = k(20). Suppose now that at an interior point I[ x( S ) II in dt J 1 Tf ~ _1_ ___1_ J 2 71 i 0 0 2 Ir 2 /I x( /) and vl'n.ich has its 8 i ii x( J0 + rei ) i9 re re iG d8. S0 + re 1·e ) JI d d8 o j 2Tr11xC S o+ ·e re1·- ) II d de, if II x( J ) ii I-= o. l/ x( J o) // d o d .e But by hypothesis f(8) = Jl x( o+ rel) II d ~ 1 and applying II x( J o) ii d 1·e the above result JI x( S0 + re ) II d = II x( j 0 ) I/ d' that is, 21f 0 S- /[ x( I ) /I d = II x( :r 0 ) JI d on r then by hypothesis JI x( r )II d = for J it is trivially true that 2.6 and hence everywhere. o in /J u c3 If II x( } 0 ) II d= 0 anct hence II x( I ) {/ d = /I x( S 0 )/1 d everywnere. Laurent's Expansion and Singularities. So far functions have been examined exclusively from the point of view of their regularity. -65- Now the influence of points at which the functions a.re not regular will be exanined. Theorem 2.6.1. o ~ R 1 r ) is a regular vector function in If x( < 1}- - 3- 0 1 < R ~ oo, then, fo r such I 2 cc xc J ) = L_ 00 a ( k - .j )n + L n n=O o n=1 an ( S- - I o) n a -n ( , t - f o )-n 2. 2 or more briefly, 00 f. x( J- ) = -eo r and < I x( 7) ( r _ j )n+1 I S - S 0 I = r, For ;my x-::- in 1'<, -:):- [ x( _?- ) J is regular in s - ~ I < 2· Consequently in this annulus 22' [ x( ~ )] = [ J - J )n +L c! - j )-n n -n n=1 n=O }"(. 0 00 00 " with b and b n -n b ( ~< [ x( t ) ] 1 = 2 11 i = 1 2 Tr i x x( J ) ] 1 ' [ powers of of ' 0 is, for instance, the circle Proof: 11.1 ~ 1 a=-n 2 II i where, J- l ~ (r - J ~ " d 1: =x:<(a), n d l" = x-3*- ( a 0 ' n=o,1,2, ••• )n+1 0 i" [ x( T ) ] (t - J \-n+1 0 b 0 J \ -n 1 , n=1,2, ••• is t hen the sum of a numerical povrnr series of ascending l 0 J 0 and a nu.111erical power series of descending powers • The corwergence of both these numerical power series in the given annulus implies t he convergence, in the annulus, of both vector power series in 2. 2, to re gular vector functions YI ( { ) and -66- x ( ~ ) respectively. Hence, ~" [ x( 2 = x-r' [ 4 ( R < 1 a n r ) + x2( J ) J ~ )] = x'~ { 4 ( S ) ] + x-:<- [ x ( ~ 2 J- for all x-';';: in 1-l:- and all ) + x 2( ~ such that I 5 - S 0 I < R2• Thus x( S ) = ~ ( l Definition 2.6.1. If R = 0 and th ere is actually at least one )• 1 lo 8 with a negative subscript in ( 2), then f point of x( l )] = S o is a singular ), namely, a pole of order m i f a -m lo 9 but an = 8 for n < -m, otherwise an (isolated) essential singularity. The behaviour of a vector f unction near a pole is contained in If the vector function x( S- Theorem 2. 6. 2. 0 < I~ - S0 / < ) is regular in I r but has a pole of order m at t x( ~ ) is not topologically bounded as -+ a -m x( Proof: = ( .r )m + • • • + r) l 0. a ~ u· --1s ) + ao + a1 ( ) - ~ s i 0 = t 0 then = 1 J [ a + a +1 ( f - f )+ ••• ·l = Cl - )m -m -m o n -J )m y( I ), 1 0 where y( } ) is a regular vector function in a neighbourhood of { = l H y( o ) = a -m f. e. y( S ) is continuous a t d for any d in D and for a -m f = S o so that I )II d = II y ( s0 ') + y( J- ) - y( r 0 ) II d ? II y( .f 0 ) II d - II y ( l ) - y( r 0 ) II d ~ /I a If _ Il a-mil d= ll a-m II d -m 2 I 2 sufficiently close to f. e there is some e in D such that II a ll lo o. -m e II x( } ) II = + • •• 0 0 and y( } ) 0 1 e lt - r 0 Im 11 y( .S- 0 • Since Then r )11 e ~ l/ a-m2JI e • Jl - S-1 olm' 0 -67- J- so that as II x( S ) JI e becomes infinite and x( ~ ) is o' not topologically bounded. Theo rem 2. 6.J. If x( } ) is a regula r vector function in r and i f ( S - ~ 0 <: bounded as )m x( l ) is topolog ically t = f 0 is a pole of order m. 0 then Pr oof : 0 t ) can be expanded as a Laurent 3eries in x( o < I~ - s 0 I < r: 00 x( s ) = n=L an ( r - j- 0) n ' -oo For any d in D, setting 1 II a -n 11 d ~ -::: 2 11 SUP r - ~L - ~ J II x( T ) JI d -n+1 0 1 1 r a=-n i 2-,, =re i G on 2 lT r ~ r r x( 1" ) --------1 d t r ( (, )-n+ r 0 , it fallows that M(d) m-n n m = M( d)r - ' for r sufficiently small using t he hypothesis of the theorem. n - m > o, a II a -n • If II -+ 0 as r-+ O, any d in D and so for n > m, d -n = e. Definition 2.6.2. A function which is re gular in the entire ~ - plane is said to be a n integral functi on. Theorem 2. 6.4. An entire function x( k ) vmich is topolo gically bounded is a constant fun:: ti on (i.e. only takes on one fixed value in 1). Proof. Since x( S ) is topolog ically bounded then for any d in D there exists M( d) x( J- > 0 such that II x( S ) I/ d ~ M( d) for all finite ) can be expand8d in a Taylor s erie s about t = 0 with infinite radius of ronvergence ( Theorem 2.5.9), x( ]- ) = f_ x(n)(O) n=O n. Sn f • - 68- The estirrBtes of Theorem 2.4.4 give for any din D I n. Lettin g r ~ oo makes the right-hand side tend to zero if n > O. for any din D, Hence I/ x(n)(O) II d = o, i .e., x(n) = e for n > 0 and so x( 1 ) = x(O) = x0 say. The following is a more g ener.91 result of the same kind. Theorem 2. 6. 5. If x( S ) is an entire function such that x( _r ) j k is to p olo gically bounded for then x( I ~ I > o, k being fixed and non-ne gative, f ) is a polynomial of degree at most k. function if k It is a consta~t < 1. The "point at infinity" in the c omplex plane may be introciuced in the usual way . of x(T) at For the behaviour of x( k ) at J = 0 will be consulted. single-valued and regular for , >- ~ = oo the behaviour r ) be To be precise let x( I t / > R. the function y( ~> I ) defined for l Setting ' t < -1 by x( ~~ f 1 =: r I ' then ) = X(-1-) = y( s I ) JI R is sincle-valued and regular there with the possible exception (as to its regularity) of the point Definition 2. 6.3. ~ I at infinity vvhi.ch y( ~ r' = 0 itself. That behaviour is assi gned to the function x( [ ) exhioi ts at ~ I ~ = o. It follows directly from the defj_nition that an integral vector func t:i. on ha.ving a pole of order m at infinity is a polynomial of degree m. Conversely a polynomial of de gr e e m > 0 has as its only singularity a pole at infinity of order m. ) -69- J As an example of this theory take 1 functions (1 9) and, let x( .f -plane to .J the S ) =e S 2 the space of integral This is a vector function on • which is regular for all finite S and The Taylor expansion of x( S ) is + ••• • Thus x( l ) has an isolated essential Now II x( } ) II m = sup J e j- z J = l zl= m 2 sup 2 e x ·l-;Y =m >- ), I( k ) are the real and imaginary parts of S • /::. o, setting x = m .) II x( 'r ) II m ?~ f )-yI( S ) where m is a positive integer, z = x + iy 2 and R( t If xR( = o. x( em n 1 ~ R( !° \ l SI 1 , y = -m I( f ) SI 1 respectively. i t follows that 1, an inequality which remains true when J ) fails to assume values whose pseudo-norms are all definitely less than one and so i-Veierstrass 1 theorem on essential singularities is not true in this theory. z x( I ) =e Y on K to _ -z z/r J is regular for all derivative x' ( S ) • e 2 1 1 z 1 + - z + - - + ••• and so x( j J2 I -T2 S f. 0 11vith Its I,aurent expansion is x( 1- = J ) has an isolated essential 2. singularity at r = 0 while ll x( k ) II m ~ m e Tfl ~ 1 showing once more that \Veierstrass theorem on essential singularities is not true in this theory. The reason for this is that in general a l ocally convex 1. t. s. is a much larc; er space than the complex numbers. may be thought of as a sequence space ( J- 1 , For instance, ~ , ••• 2 J - 70- 1 ~ l. complex and / Jn Iii -+ 0 as n-+ "" while the complex numoers form J. , namely all elements of a one-dimensional linear sub-space of the form (a ,o, ••• ,o, ••• ). 0 § 2. 2. 7. Operator Functions of a Complex Variable Hegular Operator Functions. "Z J" complete complex -space, 1' , a sequentia.i.ly complete locally convex complex l.t.s ., and I 7 (L,L) the linear system of continuous l linear functions on L into L. only the Let 1 be a sequentially Using the notation of paragraph 1.6, J b' 7 s and '7 w topolo gies will be considered. By Theo rem 1. 6.10 I J ( L,L ) is sequentially complete in the Jb topolo gy. Definition 2.7.1. U( f ) defined on function of } Let LJ be a domain in the complex plane. with values in u- I J ( L,L ) is called an operator • There are t hree notions of continuity of operator func tions available each corresponding to one of the topologies aJ b' 'J s or J w. Similarly there are three notions of ciifferentiabili ty for operator functions. These definitions V'r.i.11 not be stated explicitly since it is clear vrhat they must be. In the case of an operator function U(a) on [ a0 ,b0 ] to 7 (L,L ) the integral is obtained by considering smns of the form I n un = L U(c.) [ f(a.) - ' f(a. 1 ) ] i=1 l l lfunction of bounded vari ation in where f(a) is a numerically valued [ a ,b ] • 0 0 If these su."1ls co nverge - 71- <>-7 b ( o-:J , J ) sense to an element U in s w in the 'J b( J s , J w) inte[_;ral of U( a) and written called the b U== J I c-- J (L,L ) , then U is 0 U(a)df(a). ao 1 Theoren 2.7 •• 17:" :; converge in the '} (L,L 1 If U(a ) is v; b continuous then the sums Un J b sense and so U exists. is not necessarily complete in the ) C7f topologies and so the existence of the Ci"° J s or not ensured when U( a) is J w J 'i s and J Yf s and continuous. [/." :; w integrals is However, in the case when U( a) is re gular, the three notions of integration coincide. Jefini ti on 2. 7. 2. U( f ) is said to be re gular in t he domain of t he complex p lane if y'~- [ U( 1" ) (x) ] is regular in the classical sense for every c hoice of x in L , y'~ in (1 f~". 1 If U( S ) is regular in /:) Theorem 2. 7. 2. , then U( )" ) is J b continuous and J b differentiable (i.e. with respect to the 7 b :; (L,L ) ) in IJ any domain Proof. f) oO uniforml y with respect to Nhich is bounded and strictly interior to 1 0 f in a& • The proof proceeds along lines similar to that of It requires Lermna 2.1-~.1 applied to the function Theorem 2.4.1. y-~'c [ U( I &>':"' topolo gy on r )(x) J where x and y'~- are arbitrary elements of L and ( 1 I ) '" respectively, and also Theorem 1. 6. 11. Let r the equation be a rectif iable curve in the complex pl ane given by ~ = J (a), 0 ~ a and of b ounded variation in ~ [ o, a ] 0 a 0 where If U( t (a) is continuous t ) is a J b continuous -7 2- a operator function on r then ~ J 0 U [ S (a) ] d 5" (a) = U( I )d 0 defines a linear tra..11sformation then fr U( S )d S [ ~ U( r )d ~ J (x) has meaning and [ jr U( i )d ~] (x) = fr l U( S )(x) ] d k exists. Since In particul ar if U( S ) takes its values in U( S ) is J (L,L), T [ f U( 1 )(x) d f ] = ~ '1 [ U( S ) (x) ] r If U( ~ <J J (L,L) then eve~J =e simple closed rectifiable curve that the interior of r r in J) and such belongs to ,{) Take L I as the space of complex numbers and T = x * , an Proof. arbitrary element of L''. ·)~- for every x • 8 .1.hen Jr x-;-, [ U( f ) (x) J d ~ 0 = Hence by the corollary to Theorem 1. 6. 7 = Jr u( 1" )(x)d ~ for every x, that is, j • ) is a regular operator function on Jr u( I )d J for J~ (L, L' ) and J b continuous in Theorem 2. 7.3. to • J (L,L), where 1 is a ~ :;. -space, then, if T is in sequentially complete ~ = [ ;; U( { )d J J Jr U( S )d I (x) must be the zero element of (L,L). From here the development of t he theor;y for function on the comple x plane to 7 (L,L) proceeds as in paragraphs 2.4, 2.5 and 2.6. -73- § 3. 2. 8. Vector Functions of .::ieveral Complex Variables Vector Functions of Several Complex Variables. Much of the theory developed in § 1 of this chapter may be extended to the case of fu_rictions of several complex variables taking their values in a sequentially complete locally convex complex 1. t. s. Let Zn be the J 11 ) where S- n linear space of elements Z = ( S , ••• , 1 are comp lex numbers arid addition and scalar multiplicati on are defined by t he usual methods. z1 + Z2 = ( ~ 11,. • •' Sn1) +( S12' • • •' J z = r ( j 1' k n2) = ( f 11 + { 1 2' • • •' t n1 + r n2) • s 2' ••• ' s n) = ( 1 s 1' j j 2' ••• ' J J n). n can be made into a sequentially complete l o call r convex complex 1. t. s. n 2 (indeed a Banach s p ace) by defining II Z 11 = ~. I . Z L /l i=1 Let x( Z) = x( J 1 , ••• , J n) be a function defined on some domain, 11 that is an open connected set , 1J , of Z with values in a sequentially complete locally convex l.t.s. L. x( r 1 ' ••• ' r n ) with respect to The (strong ) partial derivative of ..\ k is defined by d x( Z) & ') k ' the limit , which is taken in the sense of strong convergence in L, being indep endent of the mann er in whi ch t he complex numbers rk approach zero. Theorem 2.8.1. Let x( ~ 1 , ••• , J n) on Z11 to L h ave first order -74- partial derivatives in each o!J- some domain (i) r k vvhere Z = ( S 1 , ••• , J- n) lies in containing the ori gin. rhen, x( f 1, ••• , J n) has partial derivatives of all orders and the TIL"ixed partials are independent of the order of differentiation, (ii) I 1 , ••• , J n) is continuous a nd topolo gically bounded x( on every closed bounded subset of /I Z II ~ r, (iii) if for II x(Z) II d II Z II and i f s < r, then for lJ , ~ ~ M(d) for each d in D, s n \I x(Z) - x(G) - L ( ~/) e \ k )/ ~ M(d) (iv) k=1 k x( ~ ' ~ , ••• , P1~0of. d II z 11 2, r(r-s) S ) is differentiable vii.th respect to J The theorem is assumed known for the s pecial case in which L is the complex plane and the proof of the theorem consists in reducin g t he g eneral case to this particular case with the aid of the continuous linear functionals x-:'c in L-i:-. (i) The nu.merical function i:" [ x( t , ••• , S n) ] ~ .S . x'" [ x( 'r 1 , ••• , :r n) J = x-i:- [ ; ~ .x ( f 1 , ••• , ~ n) ] • Hence J This implies ~ r . x-:c [ x( '.r 1 , ••• , _t n) ] J (for each fixed j) with respect to x-~ in L* and : j ~J" k for every ,;-:- in L"c. is partially ~ k since strong di ff erentiabili ty implies differentiable in each t h e weak kind. 1 x-)'c [ x( j J- k' J is partially differentiable k = 1,2, ••• ,n, for every ~ p•••, °r n)] =:'i x-i"{~ t J k ( t 1, ••• , ~ n)] j The ri ght hand si cie of this last equality says that -75- d x d~ . ( 5 1 , ... , S n) is wea.1..cly differentiable (for each fixed j) with J respect to each variable anci hence strongly diff erentiable i.e. _d____ d_ x( r 1, ••• , ~ n) exists fork= 1,2, ••• ,n. The existence d ':S k dJ j of the higher order derivatives can now be handled by an induction which is valid for every xi~ in L~~ shovrn that the order of differentiation is im..ina terial. (ii) It •Nill now be shown t hat x ( Z) is topolo gically bounded in a[} 0 , an arbitrary closed bound subset of IJ , using the k..11ovm result when the values of the function are in the complex plane. fixed x-:~ in L*, x'~ [ x(Z) J is bounded, Z in /). 0 .F'or every , i.e., x(Z) is weakly ci[} 0 • bounded and hence strongly bounded in It is now sufficient to show (iii) and this will imply the second part of (ii), nanEly x( ~ ) is continuous, and also (iv). (iii) Consider r > 0 such that II Z /I This set may therefore s erve as a IJ 0 • ~ r implies Z is in "'9 • Now by (ii) for each d in D > 0 such that 11 x(Z) 11 d ~ M( d) for Z in cf) 0 • Corresponding to each x-::- in 1-:~ there is a d in D and )1( d) > 0 s0 there ex.ists M( d) that I x':- [ x(Z) ] { ~ _,;U- (d) in ,,() 0 /I x(Z) /l ct• and by the numerical case Hence / x~~ [ x(Z) ]{~fl (d)M(d) -76- U n J x* [ x( Z) ] - x* [ x( 8) ] - k=1 =}x~:- [ x(Z) -x( 8) - k /I Z I/ f s ~ r. d k o !I L c2...!) k=1 k J M(d) r( r-s) Applying Theorem 1. 6. 7 this gives n x(Z) - x(e) - j1 k t ("d() ~ Jl ~(d) k=1 for xl' [ x 1 J [ Ci Sk " }. ~ 2 II ~ M(d) II Z I/ • k d r ( r-s ) JI Z 11 2 -77- Chapter III Functions on Vectors to Vectors 3.1. In this chapter, functions on one sequentially Introduction. complete locally convex complex 1. t. s. to a second such space will be studied and in particular functions which are analytic in a sense to be specified later. duced in § 1 The necessary concepts of differentiability are intro- and the properties of differentiable functions are studied The properties of analytic functions are treated in § 2 while there. in § 3 the analogue of the Cauchy-Riemann theory of functions of a complex variable is discussed. § 1 Differentiable Functions 3.2. Gateaux Differentiability. In this section, unless other- I wise stated explicitly, Land L will always denote sequentially complete I locally convex complex l.t.s . an:i D and D the correspondi~ associated directed systems. Definition 3.2.1. Let f(x) be a function defined on some open I set in L and taking its values in L. Suppose that for x 0 in and each y in L, _()_ f(x0 + r y) - f(xo) ~f(x ;y) = lim - -- - - - - - o exists, where (i) S-+0 is complex. J Then f (x) is said to be G-diff erentiable at x S', f (x0 ;y) = =x 0 and &f(x ;y) is called the G-differential of f(x) at x = x 0 0 -78- with increment y, (ii) f(x) is said to be G-differentiable in G-differentiable at each point of (iii) Jl_ i f it is ft , the G-differential ~ nf(x;y1 ; ••• ;yn) is defined by ~ n+1 f ( x;y1 ;y2; ••• ;yn+1 ) = S0 f(x;y1 ;y ; ••• ;yn) 2 = (iv) S nf(x;y) = f(x), and d nf(x;y;y; ••• ;y ) • In much of this theory the domain of the fun:: tions considered is i1rnnaterial while their range can be extended to linear topological spaces. Thus almost all the theorems of the Banach space case are valid (2,21). The notion of G-differential is closely related to the notion of derivative of a vector function given in Definition 2.2.3. Theorem 3.2.1. f(x) defined on the open set ' ' and y in L f(x + S y) is a regular function of J y is in .fl. Proof. - whenever . _9._ f(x + 1 y) = l im -1 [ f(x + ( J + 7: )y - f(x + i y) ] 1 [f(Qc + l y) + i- y) - f(x + ~ y)] d~ t ..... 0 t = lim t ..... 0 t = In particular, in L to if and only if for every x in .IL L is G-diff erentiable in x + JL 6 f(x + ! y;y) -79- r~ f(x + i y~ ll1 =O ld ~ Theorem 3. 2. 2. 11.. . .fl ~ nyn) is partially differentiable with respect Proof. =-r ~lmo if Sk' k =1,2, ••• ,n, whenever x + J 1y1 + • • • + J ny n is in to the _()_ 6 f(x;y). f(x) is G-differentiable in the open set and only if for every x in f(x + S 1y 1 + ••• + = i k [ x + t 1Y1 + • • • + ( j k + 7: k) Yk + • • • + SnYn) f ( f (x + $' 1Y1 + • • • + = lim } l k-+ 0 [f ( x + k 1" ~k + • • • + J- nyn) ] 51y 1 + • • • { k1k + • • • + t nYn) + 'L ~k) - In particular, o,o Theorem 3.2.3. open set _()_ = ~ f(x;y. ), 1. = 1,2. Let f(x) be defined and G-differentiable on the in 1 with values in 1 1 • Then, for every yin L, ~ f(x;y) is a G-differentiable function of x in Proof. i JL • -80- Let 0(x) = 6 f(x;y1 ). Then, 3 0(x;y2) =[d~ 2 0(x + ! ;i\'2l] j 2= 0 ' and hence, By Theorem 3.2.2 f(x + 1 1y + ~ if ) is a partially differentiable 1 2 ~ function with respect to 1 and l 2 and Theorem 2. 8. 1 ensures the existence of higher derivatives. The orem 3.2.4. open set ( i) Jl in L to 1 ' • [a [---C ) _ n __ d.r1.... Q) n J1 •••• (ii) Let f(x) be defined and G-differentiable on the Then n f(x + [ .k=1 ~S' n C> 's f(x + I !2'kl ] n r k :::() = ~ f(x;y1; •••• ;yn)' £. k=1 n. = ~ nf(x;y). Proof. The result (i) will be proved by induction. For n it is true by Theorem 3.2.1 and for n = 2 by Theorem 3.2.3. that (i) is true for all values of n up ton -1. =0(x). Then, a 0(x;yn ) = [ ~ 0(x + dS n Set =1 Suppose now ~ n-1 f(x;y1 ; ••• ;yn_1 ) -81- that is, }nyn) + .1"1y1+ ... + tn-1Yn-1~ J o, ••• ,o 0 • o,o, ••• ,o (ii) is an immediate consequence of (i) and Theorem 3.2.1. Theorem 3.2.5. T L • Let f(x) be defined on the open set _f1_ in L to t Then G f(x;y) is a linear fu..~ction on L into L • Proof. It is clear that ~ f(x; ~ y) = 1 ~ f(x;y). now the .function f(x + f 1y1 + S ~ 2 ) of ( Consider f 1' S 2). By Theorem 2.8.1, t I for any d in D I where K(d ) is a real number. 1 By Theorem 3.2.2 this gives llf(x +~ 1 y + .\"~ )-f(x)- 1 1 ~f(x;y )- 2 1 11 c ~ 1 , Letting t1 = f 2 = ~ llf(x+ ~ (y1 -fY'i)-f(x)- ! 2 .ff(x;y2)lf d' ~ K(d') s 2) II 2• this last inequality now reads ! [ sf(x;y1 )+ r f(x;y 2)] II I d ~ ' K( d )2 I J I2 -82- or, Hence, Theorem 3. 2.6. to L' • Let f(x) be defined on the open set SL in L ~ nf( x;y1 ;y ; ••• ;yn) is a completely symmetric nrultilinear 2 form in y1 ,y2, ••• ,yn which is G-differentiable with respect to x when x Then is in _fl ~ nf(x; ,.\ y) = ). n ~ nf(x;y). • By Theorem 3.2.4 (i) Proof. f nf(x;y1;y ; ••• ;yn) is symmetric in 2 is linear in the last argument (Theorem 3. 2.5) implies that it is linear in each yk. The G-differentiability follows from Theorem 3.2.3. 3.3 Series Expansions of G-differentiable Functions. Using the relations between G-differentials and derivatives the expansion theorem for analytic vector functions of a complex variable can be applied to obtain a series expansion for G-diff erentiable functions. Theorem 3.3.1. open set _[)_ Let f(x) be defined and G-clifferentiable on an in L to L'. Let x be in 0 fl . Then f ( x) may be expanded about the point x in the form 0 $ nf(x0 ;x-x0 ) ~ f(x ;x-x ) 0 0 f(x) = f(x0 ) + ----:-,-+ • • • + ---.,,..,-- + ••• 1. n • ' -83- the series being convergent (at least) for all x in any neighbourhood fl contained in about x • 0 Let N(x ;d,r ) 0 Let 0 < f [ x in 0 < r ..n I II x-x II d < r 0 and let x be chosen so that 0 I\ x-x II 0 0 d , d in D ] ~ e <.. r 0 • ro If r is now taken satisfying the inequality 1 < r < - • , then by €- Theorem 3. 2.1 the function f(x 0 + j y), y = x - x0 , is analytic for l l' I < r and by Theorem 2. 5. 9 can be expanded (for I S I f (x + .t y) 0 = Lf- [ - dn n=O f ( x + o< y) d ~n In particular for ~ nf(xo;y) 00 = f(x0 + x - x0 ) = [ n •' n=O Fran Theorem 2.4.4, where r n ' • o( =O using Theorem 3.2.4 (ii). f(x) I n ] 0 S nf(x0 ;y) -is any circle - if- $nf(xo;y) - L , n=O n. =L. &nf(x ;x-x ) n=O f(xo + 2 Tf i Ij I =f < >- n .l .t = 1, 00 lr n : < r) as } y) J n+l 0 0 n • d~ • ' r, and in particular may be taken to be of unit radius. If for all x in N(x ;d,r ), f(x) is topologically bounded, that 0 0 is, its values for such x form a bounded set in L ' , then for every d in D' there is a number M( d ' ) 11 r Cx) II , ~ d ' M(d ) > 0 such that and M(d ' )n •t ' -84- In general while it is impossible to make any for x in N(x ;d,r ). 0 0 statement about the topological boundedness of f(x), the following is a step in that direction. Theorem 3.3. 2. Suppose f(x) with values in L G-differentiable on an open set .IL ' is defined and of L where e is in J1 . If N(8;d,r ) is determined as in Theorem 3.3.1 and if S is a subset of 0 N(8;d,r ) which is contained in a finite, say n-dimensional linear 0 subspace Ln of L and compact there, then f(x) is bounded in S and the 00 L ~ n f( e;x) n=O n • where 0 < yt < 1. series converges uniformly in s1 ' Further, for each d = Sn N(8;d, "1 r 0 ) ' in D' , the series 00 \' L ll$nf( e;x) ll d' , n=O n. Proof. converges unif onnly in s • 1 Ln is homeomorphic to Zn (see paragraph 2. 8) . . . n} '-n = Ln n N( 8; d ,r 0 ) • tains 8. Every x in Let .fl n is an open connected set in Ln and con- __n_n is of the form x = n where ~, ••• ,xn is a fixed basis of L } 1 ~ + ••• + f nxn and the j . are complex J. j 1, • • •, j n) fonns a homeomorphic nu.rnbers. The set cl> of all. these ( image of ...D... n in Zn containing the origin in Zn, that is,~ is a domain in Zn containing the ori gin. f( s 1~ + ••• + J- nxn) is a partially differentiable function with respect to J 2' • • •' :S- n ~ for (Theorem 3.2.2). ~ 1, Now S C n ) in J) f l n is a compact set in Ln and so its -85- homeomorphic image, which is contained inc», is compact in Zn and By Theorem 2.8.1 (ii), the values of hence closed and bounded there. f(x) then lie in a bounded set for x in s, that is, given d ' in D' there I is M(d ) > O such that I II f ( x) II d 1 < M(d ) for all x in s. 1 Choose r such that 1 < r < - , where 0 < I = 2~ i r where 1 Jr f( n. , M( d') rn • S x) = 0 and II x II d ~ "1 r 0 • • f r 0 ) and d ' in D' Consequently for x in s1 = S n N(8;d, 5- Then j n+1 is a circle of radius r about ffd'nf(e ,x) II d' ---,--- 1. Since r > 1 , j ~ L., is convergent n=O and hence ex> \ Li n=O llinf(G,x) __..,,,..........__ll d' n. converges unif orrnly for all x in s • 1 The expansion of Theorem 3.3.1 is valid in a larger region than the neighbourhoods considered there. This larger region exploits fully the character of the G-differential as a directional derivative. Definition 3.3.1 (2). about x 0 if c-r'"(x ) = x 0 0 ><. The set c"(x ) c 0 + X, where x in X and L is called a c-star I ~I ~ 1 implies J- x is in X. Definition 3.3.2(2). If J1 is an open set in L, then for x in _fl , r(x,y) will denote the suprennun of all numbers r such that I) I ~ r implies x + J y is in Jl . -86- If Jl C L is open and x about x , the c-star in - fl about x • 0 actually a c-star is easily verified. fl . implies x + j- y is in x 0 fl + ( :S $ )y is in SL • then the . set of all points S' y is in fl I 5 I ..::: 1 implies x + x0 + y for which 0 is in 0 That this set of points is Let y in L be such that I ~ I ~ 1 then If is a c-star I ~ I~ 1 I 's" ~ I ~ 1. Hence In particular a neighbourhood about x 0 is certainly a c-star about x • 0 set Let f(x) on L to 1 ' be defined and G-differentiable on an open JL • fl , then for IS I < If x is in 0 r(x ,x-x ) the Taylor 0 0 expansion (Theorem 2.5.9) ao f(x o + s (x-x o )) = [ r ~ f(x + do<' n o o( n=O (x-x ))] o --r o( =O n. n ao =[ S nf(x ;x-x ) o o n=O .L = n' • (X) Snf (x l (x-x ) ) =L n=O ' n • 0 ; 0 holds . The point x 0 is arbitrary. If S (x-x0 ) is replaced by (x-x0 ) then this may be stated as Theorem 3.3.3. __n For x 0 in fl t ao f(x) =[ n=O Snf(xo;x-xo) t n. • and x in the c-star about x 0 in -87- Theorem 3.3.4. [ fn(x) J If is a sequence of G-diff erentiable functions on L to L' defined and G-diff erentiable on an open set and if and [ fn(x) ~ Jl. converges uniformly on the 11 discs 11 x + .S- y, y in L I s L ~ r' < r(x,y), of fl differentiable in J1 . , then the limit will also be G- Further, under these conditions for x in JL , yin L, S kf(x;y) = lim S k f n (x;y). n ~co Proof. The proof of both these statements is a consequence of Theorem 2.5.10. If f(x) = lim fn(x) it must be shown that for x in n ~co fl and y in L, f(x + i y) is differentiable with respect to for I ~I < r(x,y). Then by Theorem 3. 2.1 it will follow that f(x) is G-differentiable in .JL • of analytic fumtions for I 's I ~ But { fn(x + 1 y) } being a sequence IS I < r(x,y) converging uniformly for r' then by Theorem 2.5.1 0, the limit function f(x + l y) is IS I < r(x,y). an analytic vector function of a complex variable for 3.4. J- F-differentiability and Partial Differentials. The first non-metrical definition of a differential (the M-differential) for functions whose arguments are in a 1. t. s . and whose values are in a second (possibly distinct) l . t.s . was given by A. D. Michal in (22). When the 1. t . s. are normable spaces ever-J Frechet differentiable function is also an M-differentiable function but not conversezy. In ( 23) however, Michal defined a 11 first order diff erentialll for functions -88- f(x) with arguments and values in topological abelian groups with the property that if the topological abelian groups are Banach spaces then f(x) is Frechet differentiable and the two differentials are equal. Even earlier in (12) and (1J) Michal and Paxson had defined a differential for functions on a l.t.s. to the same l.t.s. but this - differential did not have the composition property. A comprehensive account (with extensive bibliography) of these matters, that is, of topological differential calculus, appears in (24). The differential used here is due to Hyers (25) and is a differential of the type defined in (22). Further, in Banach spaces, this differential reduces to the F'rechet differential. Definition J.4.1. The furotion f(x) defined on an open set ' will be said to be F-differentiable at the point x 0 of S2. with increment y in L and df(x0 ;y) = d f(x 0 ;y) will denote its Jl c L to L I F-differential if (i) df (x0 ;y) is linear and continuous in y on L to 1 ' , ' in D' , there corresponds d in D with the property that given '>'"( ' > 0 there exists ~ > 0 such that (ii) for every d II f(x + y)-f(x )-df(x ;y) II 1 ~ Yf ' II y II 0 Here d 0 0 = d(d ' ) and r = F-differentiable in JL d ~ (d,d ' , 7/ ' ). whenever llyll ~ f • d d said to be f(x) will be if it is F-differentiable at each point of -89- dnf(x;y; ••• ;y) will be written as dnf(x;y). Theorem 3.4.1. If f (x) and f (x) are F-differentiable at x =x , 0 1 2 = ~ 1f 1 (x) + ! f 2(x) is F-differentiable at x = x and then f (x) 3 0 df)x ;y) 0 Theorem 3.4. 2. open set _Q open set in J2 fl S 2df 2 (x0 ;y). If f(x) is defined and F-differentiable on an in 1 to 1 Theorem 3. 4.3. = 51 df1 (x0 ;y) + ' then f(x) is continuous in _f2 • If f(x) is defined and F-differentiable on an in 1 with values in 1', then f(x) is G-differentiable and the two differentials are equal. Thus the F-differential when it exists is unique. Theorem 3.4.4. ' with values in L • Let f(x) be defined on an open set __n_ of L Let D,E be two associated directed systems for 1 defining equivalent systems of pseudo-norms for 1 and D' , E f be two associated directed systems for L ' pseudo-norms for 1 • I defining equivalent systems of If f(x) is F-differentiable at x 0 with respect 1 to D and D then it is F-diff erentiable there with respect to E and I E and the F-differentials are the same. The proofs of these theorems are straightforward and will not be given here but the methods used are illustrated in the following theorem. Theorem 3.4.5. Let L , L , L be locally convex l.t.s. with 1 2 3 associated directed systems n , D , n respectively. 1 2 3 Let f(x) be defined on the open set _f2 fl 2 c 1 c L to the open set 1 L 2 -90- Jl. 2 to Lr and let g(y) be defined on at x =x If f(x) has an F-differential in G and g(y) has an F-differential at y = y = f(x) in G , 1 2 0 0 0 then h(x) = g(f(x)) has an F-differential at x = x = dg(f(x0 ); df(x0 ;x)). dh(x ;x) 0 Proof. ~ given by 0 The proof of the theoran consists in showing that given 17 3 in n there exists d1 = di(~) in D1 such that for any 3 ~ 1 = there is ~ 1(d3'di, ""{ 3) 0 so that II x II ~ < ) II h(x + x)-h(x )-dg(f(x ) ;df(x ;x)) II 0 0 0 0 Corresponding to ~ there exists d Y( t > o, J. implies 2 d3 ~ Y} 13 ~ f 3' II y II d3 Since f(x) is differentiable at x is in D and 1 0 o( = • II y II d < S 2 2 • it is continuous there (Theorem J. 4. 2). 0 d 2 o< (a ) ) 2 ~ 1 implies d2 II f(x + x)-f(x ) II < ~ 2 for II x II 0 ~ o, = d (d ) in n such that for 2 2 3 ~ 2 = S2(d ,d2, 7 ') can be found so that 3 3 II g(yo + y)-g(yo)-dg(yo;y) II Hence II x II > O. ~ < o< , where e1 = e1 (d2 ) Taking y = f(x + x)-f(x ) it 0 0 follows that II g(f(x +x) )-g(f(x ) )-dg(f (x ) ;f(x + x)-f(x ) ) II ~ n3 ' II f(x + x)-f(x ) II 0 for 0 0 0 0 ~ { 0 II x II < o< • Then, using this last inequality, the properties 0 e1 of the pseudo-norm and the linearity and continuity (as a function of y) d2 -91- II g(f(x0 + x) )-g(f(x0 ) )-dg(f(x0 ) ;df(x0 ;x)) /I d._J ~ V ' II f(x + x)-f(x ) II d + II dg(f(x ) ;f(x + x)-f(x )-df(x ;x)) II d... 3. 0 2 0 0 0 0 3 0 ~ 13' II f(x0 + x)-f(xo)ll d 2 + M(~,e 2 ) ii f(x0 + x)-f(x0 )-df(x0 ;x)lf e ' 2 < o<. • Since f(x) is differentiable at x = x 0 , corree1 sponding to d and e there exists b = b (d ) in D and c = c (e ) 1 2 1 2 1 2 1 1 2 for II x II in D1 such that given 1" = '? 2 > 0 there are (3 = (3 (d 2, b1 , r; 2) , "(' (e ,c1 , { ) respectively so that 2 2 12 II x II b + M( d 2,11 )llx II ~ II f(x0 + x)-f(x0 ) II d ~ ~ 2 II x II b + II df (x0 ;x )fld ~ where II xii b < ~ 1 k.i (d2) is in D1 and M(d2,11) > o, and 2 , ~= 1 II f(x O+ x)-f(x0 )-df(x0 ;x)ll e for II x II c1 < r . 2 ~ 2 '1 (2 llx II 1 C1 These last inequalities then give II h(x0 + x)-h(x0 )-dg(f(x0 ) ;df(x0 ;x) II d3 ~ ~ 3 ' ( 72 11 x II b for II x 11 c1 e1 < o( , 1 + M(d 2 ,k.i) II xii 11 ) + M(d3'e 2 ) 72 I/xii C1 , II x 11 b < R and II x II < 1"' 1 1c1 o • Tald.ng ~ ) e1 , b 1 , ,11 ,'1 in D1 , then since these last depend on ~ so does ~. )3 ' , /2 can be detennined so that 1J' ( 12 + M( d2' 11 )) + M( d3' e2) f 2 < 1 y Finally if S 1 =min( o<. , / , y ), then for II xii di< ~ 1 II h(x0 + x)-h(x 0 )-dg(f(x0 ) ;df(x0 ;x)) II ~ ~ ~ 3 /Ix fl di. -92- It is clear that dgf(x ;df(x ;x)) is a continuous linear function x 0 0 and hence it is the F-differential of h(x) at x = x0 with x as increment. Let L(R) and 1 ' (R) be locally convex real l.t.s. and L(C), 1 ' (C) and L(K) and 1 ' (K) the associated couple and composite spaces respectively. If f(x,y) is defined on an open set in L(K) to 1 ' (R) partial G- and partial F-differentials with respect to the variables x and y can be defined. · Thus the partial G-differential of f(x,y) with respect to x is the ordinary G-differential with respect to x of f(x,y) when y is held fixed in L(R). By the total F-differential of f(x,y) is meant the F-differential of f(x,y) with respect to the composite variable (x,y). It is written df(x,y; ( ( Ax, A y) is the (composite) increment. l:l. x, Ay)) where This is to be contrasted with the F-differential dg(z; A z) of a function g(z) defined on an open set of L(C) to 1 ' (C). z = (x,y) and In this last z is the oouple element Cl z is the couple increment. § 2. J.5. Polynomials. Analytic Functions As before L and L' will denote sequentially complete locally convex complex l.t.s. and D and D' the corresponding associated directed systems. Polynomials on the complex numbers to a sequentially complete locally convex l.t.s. have already been defined in Definition 2.5.2. -93- Definition 3. 5.1. A function p(x) ::/= 8 on L to L' is called a polynomial if (i) p(x) is continuous at every x, (ii) there exists an integer n such that for any pair (x,y), x,y in L, p(x + r y) is a polynomial on the complex numbers to L ' • The least number satisfying (ii) is the degree of p(x). A polynomial p(x), homogeneous of degree n, that is one for which p( J- x) = Jnp( x) is a polynomial of degree n. Hence the phrase homogeneous polynomial of degree n is used to describe such polynomials. Definition 3. 5. 2. be an element of L. Let f (x) be a function on L to 1 Then ~ yf(x) t and let y =f(x + y)-f(x) is called the first difference of f(x) with respect toy. If x,,x 2, ••• ,xn are elements of L, then the nth difference of f(x) with respect to x,,x 2, ••• ,xn is defined to be f(x)) Definition 3.5.3. ' non L to L. Let h(x) be a homogeneous polynomial of degree Then the function h(x,,x2, ••• ,xn) on LxLx••• xL to L' defined by n.' -94- is the polar of h(x) with respect to Xi ,x 2, ••• ,xn. The algebraic properties of polynomials, homogeneous polynomials and polars are treated at length in (2,26). With regard to properties involving the topology of the space L and 1 ' certain changes from the Banach space case are in order. Theorem J.5.1. on 1 to L' • Let h(x) be a homogeneous polynomial of degree n Then given d ' in D' , there exists d in D and a least t M(d,d ) > 0 such that 11 h(x) II d, Proof. ' II x II nd , all x in L. M( d,d) ~ By definition h(x) is continuous at x = e. It follows that given d t in D ' , there exists d in D and ~ > 0 such that II x 1J d < b II h(x) II d' ~ 1. Let x be any element of L implies such that x /= 8 or ll x II d /= o. Letting x = f2 • llx IIx d , then and consequently II h(x) 11 , d that is, 11 h(x) II , d ~ n II x II d • The inequality is trivially true when x = e or II x II d = o. This theorem makes possible the following definition. -95- 1 Definition J.5.4. If d and d satisfy the statement of Theorem 3.5.1 then sup 11 h(x) II d' exists and is denoted by ~(d,d , ). x in L II x II~11 x ll d/= O I t t 1\i(d,d ) is called the modulus of h(x) with respect to d and d • If h(~ , ••• ,xn) is the polar of the homogeneous polynomial h(x) then it is a linear homogeneous polynomial in each of the arguments Further h(x) = h(x, ••• ,x). and is symmetric in the arguments. Theorem 3.5.2. Let h(x) be a homogeneous po~omial of degree n t on L to L and let h(~,x 2 , ••• ,xn) be its polar. Then h(~, ••• ,~) is continuous at (e, ••• ,e) and hence bounded with respect to the pseudo-norms. Proof. is of degree zero in x so that Let ~ , ••• ,xn be n and arbitrary set of increments and take x = - ~ L i=1 the successive differences of h(x) gives rise to 1 n L h(-2 • E .x.)' 1 '1 1=1 By Theorem 3.5.2 given d t t 1 in D there exists din D and M(d,d ) > such that II h(~ 1 n n L E . x. ) II i=1 1 1 t d ~ M(d,d')( IJ ~ ~ M(d,d ' ) nn (max II xi II )n. 2n i d c. [ i=1 0 n f ' \ n M(d,d >c L llx. II ) .x. 11 )n ~ 1 l. d ~ i=1 l. d -96- Consequent]y II h(~ ,x2, ••• ,x ) 11 n 1 -, ~ d' n. M(d,d ' ) rnn n. (max II x. II )n i ]. which can be made arbitrarily small for max II x . II i l. d d sufficiently small. Thus h(~ , ••• ,xn) is bounded with respect to the pseudo-norms (Theorem 1.7.1). This theorem makes possible the following definition. Definition 3.5.5. sup x.in L ]. By theorem 3.5.2 d ' and d exist such that 11 h(x.. ,x , ••• ,x ) II , 2 I n d t = Mn(d,d ) is finite. 11 xi II d~ o called the modulus of the polar h(~,x , ••• ,xn) of the homogeneous 2 polynomial h(x) of degree n with respect to d and d ' • Theorem3.5.3. (d, d '_) be the modulus of the homogeneous Let1~ polynomial h(x) of degree n on L to L ' • polar exists and conversely. ' The modulus M (d,d ) of its n Further • Proof. Taking II x. II l. d = 1, i = 1, 2, ••• ,n it follows from the last inequality in Theorem J.5.2 that ' nn ~(d,d ) ' n. • -97- I Then, Mn(d,d) = II h(x-px2, ••• ,x) I\ , f: sup II x. 11 1 d = 1 n d ' II h(Xi ,x2' • • • ,xn) II d' Finally, M ( d,d ) ;: sup n II Xi 11 II x 11 II x II 2 d••• d n ~ sup 11 h(x,x, ••• ,x) II d' d II x ii~ =~(d,d ' ) 3.6. The Series Expansion for Analytic Functions. In the remainder of this chapter the word domain will be used, as usual, to designate an open connected set. In a locally convex space the notions of connected and arc-wise connected are equivalent. Definition 3.6.1. domain of) A function f(x) on 1 to 1 ' defined in the continuous and G-differentiable in Theorem 3. 6~. (i) JJ is said to be analytic in 1J if it is single-valued, • If f (x) on 1 to L' is analytic in f(x) is F-differentiable in }) iJ , then and has F-differentials dnf (x;Xi ;x ; ••• ;xn) of all orders ·which are analytic functions of x 2 in JJ for fixed Cx, ,x , ••• ,x ) and jointly continuous in the 2 n increments (~,x , ••• ,xn) for fixed x, 2 (ii) the series expansion co f(x + y) o is valid for every x lying entirely in 0 11 =I: n=O in • .id n. and x + y in any neighbourhood of x 0 0 = -98- Proof. By Theorem 3.2.6, since f(x) is G-d.ifferentiable S nf(x;x, ;x ; ••• ;xn) exists for all n and is a cornplete:cy- symmetric 2 multilinear form in the increments x,,x , ••• ,xn. 2 f(x) may be expanded as a Ta;y-lor Series of G-differentials about any point x in ,,!J (Theorem 3.3.1). E Let x be an element of 0 0 o(J • Given d ' in D' and ' > o, there exists d in D and '7 > O such that implies II f(x)-f(x ) II II x-x0 II d < -¥ • < d' Then for E '. II x-x II 0 d < 7 Pick any point x such that 0 ~ and such an x it is true that II y II d ~ co \ ~ nf(x;y) f(x + y) = L , n=O n. r where , being the circr e with unit radius. < t: ' and so II f (x+ S y)-f(x ) II then 0 =M(x ,d ' , E ' ). 0 d' Since II x + ~ y-x 0 r nf(:x:; 1 2 d <. '7 , II f(x+ s y) II , < t: ' + II f(x ) II d 0 Thus, Let y be any elenent of 1 such that y -/= e and II y II d -/= o. II II y llylld ) JI ' ' n • M(x0 ,d . , d' , , n • M(x0 ,d ' Then, t. ' ) , or , 2 n n E ) (-) II y JI , 1 d 1l II x-x0 II < ~ • This same inequality is trivially true when y = 8 or JI y II d this were not the case then a d c. = o. If > 0 could be determined so that d ' -99- ll~nf(x;ay) 11 d ,= an II S nf(x;y) II , > n~ M(x0 ,d', E ') wbile d n 11 ay II d = o, II x-x0 lld < ~ , giving a contradiction. af(x;y) Thus as a (linear) function of y is bounded with respect to the pseudonorms and hence (Theorem 1. 6. 1) is continuous :in y, locally uniformly in x. This same inequality combined with the Taylor expansion implies ~ . that for given II f(x+y)-f(x)- > O, ~ f(x;y) II , ~ d 4.M(x0 ,d ' , E ' ) /I y I/ d 2 I </ -2 11y11d) t :5 when II x-x 0 II d < ; 1 llylld ' . ( /1 and 11 y II d < min "2 , , n 2 11' l I , 2 { + 4M(x0 ,d , ) • 1 € ) With this last relation the proof that f(x) is F-differentiable with df(x;y) = S f(x;y) is complete. It will now be shown that under the hypothesis of the theorem S nf(x;y) is an analytic function of x for fixed y. that Theoran 3. 2.6, Since by 5 nf (x;y) is G-differentiable it remains only to prove the continuity of Lemma 3.6.1. S'nf(x;y) with respect to x. This requires Let f(x,y) be a function on L x L 1 2 to a space L where L , L , L are sequentially complete locally convex complex 1 2 3 l.t.s. Suppose f(x,y) is defined and contirmous in the pair (x,y) 3 -100- for x in a domain if x 0 is in l) .[) of L1 and y in a compact set G of L , then 2 , f(x ,y) is continuous at x , uniformly with respect 0 0 to y in G. Then, given d in Dy 3 > o, there corresponds to any neighbourhood N(x0 ) of x0 at least Proof. ~ 3 Suppose the theorem is false. one x in N(x0 ) and at least one y in G such that II f(x,y)-f(x0 ,y) II cJ > ~ y f(x0 ,y) being jointly continuous for x = x 0 and any y in G, then for each pair (xo ,y), yin G, there are neighbourhoods N(x) in IT.. , N(y) o in 1 2 such that x f in N(x0 ), y ' in N(y) implies II f(x ,y ) -f(x0 ,y) ll < ~. 1 1 d 2 3 These neighbourhoods N(y) of y, for all yin G, constitute an open covering of G. By compactness there is a finite number of them N(y1 ), N(y 2), ••• ,N(yn)' say, constituting a covering of G. Let N1 (x0 ), N2(x0 ), ••• ,Nn(x0 ) be the corresponding neighbourhoods (in L1 ) n of x 0 • There exists N(x0 ) C . 1 Ni(x0 ) • The continuity condition n 1=1 then says that for x in N(x ), yin N(y.), 0 1 c\ 3 II f(x,y)-f(x0 ,yi) II ~ < 2 . Corresponding to this same neighbourhood N(x0 ) of x 0 there is some x in N(x ) and y in G such that 0 II f(x,y)-f(x ,y) II 0 ~ > r 3. 3.1 -101- On the other handy must lie in some neighbourhood N(y.), j J = 1,2, ••• ,n. Calling it N(y ) then for x in N(x ) , y in N(y ) it is true that 0 0 0 ~, It f(x 0 ,y0 )-f(x0 ,y) II d3 < and ~ and these contradict the inequality 3.1. With this it can be shovm that Snf(x;y) is a continuous function of x, for fixed y. Suppose that x and let y be an is in 0 Using Theorem 3.3.1 an element d in D arbitrary fixed element of L. and positive numbers r ,r can be found so that x + S y is in I) when 0 11 x-x II 0 d I ! l < r and with these restrictions < r 0 and n: S nf(x;y) = 2 lT i r being the circle I~I = I'< r. 1 all since ~ of r 7 ~ ! n n+f ~ f (x;y) is continuous at x From this it follows that II x-x 0 II e < ' in D , f(x+ 's y)-f(x + ~ y) II 21T E ' t n+T Then given d 1 that for a given f(x+ ~ y) r ' >0 II 1 I dJI d' = x0 provided > o, e in D and '7 > 0 can be picked so that implies II f(x+ J y)-f(x0 + ! y) II d, < ~ for r . This can be done as a consequence of Lemma 3.6.1 is compact. Thus for II x-x 0 < II d r0' /J x-x0 < JI e 7, -102- 6 nf(x;y) considered as a function of x for fixed y is analytic and in )) • Applying the argument vbich was used to show that the analytic function f (x) is F-differentiable it follows that the analytic function ~ nf(x;y) is F-differentiable with respect to x in ~ • But the same conclusion then extends to is the polar form of the homogeneous polynomial d nf(x;y) and hence linearly expressible in terms of the functions ~ nf(x; E 1x.i+••• + fnxn) ' E . = o, 1, all of which have just been shoNn to be continuous Fi differentiable functions of x. Further, as in Theorem 3.5.2, S nf(x;.x.i; ••• ;xn) is jointly continuous in (.xpx2, ••• ,xn). The b nf(x;y) to obtain the notation dnf(x;y) may be used in place of expansion co n. Theorem 3.6.2. • t A homogeneous polynomial p(x) on 1 to L is an analytic function in 1 and hence it is F-differentiable. Let fR be a sequentially complete extended locally convex complex topological linear ring with associated directed system D (Definition 1.8.J). Let I? be commutative and have a unit element u. 2 Definition 3.6.2. exp x = u + n T+ ~2. + ••• +n.-' + ••• , x n o where x =:xx ••• x, a product of n factora and x = u. -103- As stated in paragraph 1.8 part of the motivation for introducing extended locally convex complex topological linear rings is to ensure the existence of entire functions (in particular the exponential function) on these rings. If the ring is not extended it may not be possible to define an exponential function (14). Theorem 3.6.3. For each x in I?.. , the sequence of partial sums of exp x converges and defines an analytic function on tR. into itself. E > o, d in D, let x be any element in n xm+1 x + - I + ••• + (m+1). n.' ll x ll ~ + II- x ii m+ d + ••• + 1!...?!J.lI (m+1 )! n. m.' Pro of: Given &?. • Then J/ ~ for n > m ~ and since ti< m (d, f ). 0 1 II d ~ a < (- , Hence the sequence of partial sums converges is sequentially complete, converges to an element of • Further exp x is a continuous function of x. ~ > 0 co co exp(x + y) - exp(x) = I: (x n.+ y)n n=O co = L ~ E = 1 + ll X ltd e - 1 L -,xn. n n=O (x + y)n - xn n.' n=O Picking For, given d in D, < • 1 it follows that for 11 y II d .:'... ~ , -104- co n II exp(x + y) - exp(x) II d = II [ -n.' [ 1 n-v yv II ... n L [ b -, n=1 • d )J =1 n=1 < (~) x (~) =1 n- v II x/l d n. y ~ n [ (1 + II x 11 d)n - II x ii d co r = n=1 n •' [ f .., ~ s ] (1 + ll x ll d)n n. n=1 c 1 + II x /I d (} (e - 1) = E • = 00 exp x is also G-diff erentiable. exp(x + t y) - e:xp x = lim J J -+ 0 QI) In=O (x + T y)n n.' L -L1 n=O I n. 00 00 I 1 .) -+ 0 n=1 n. =lim (x + _r y)n _ xn f :r L = n=1 (x + } i'.)n - xn llm -1n.' J-+ t 0 00 "" x =[ -1 nyxn-1 =y L , = y exp x. n=1 n.' n=1 (n-1 ). Thus n-1 S exp(x;y) = y exp x and the F-differential d exp(x;y) =y exp x. 3.7 Further Properties of Analytic Functions. Theorem 3. 7.1. Let f (x) be an analytic function on 1 to 1 f vanishes in some neighbourhood in its domain of analyticity ~ • f(x) vanishes identically in Proof. which Then otJ • f(x) = 9 for all x in N(x ,d, c ) 0 [x I II x-x0 II d < E ] , -105- where N(x ,d, E ) C 0 o8 • The expression for ~ nf(x ;x-x ) = e for integral shoos that 0 0 II x-x I\ 0 be any element of L such that x - x & nf(x ;x-x ) as an 0 d 0 <f Let x • I= e and II x - x 0 II d I= o. 0 Setting x -x f y=2 x-x 0 ll x-x II 0 + xo,, then 11 = d 11 % II y-x 0 d ll x-x 0 II 0 d E =- < E2 /I ,, d and consequently x -x E _ _o_) = (-) 2 11 x-x 11 0 that is trivially true. 0 any neighbourhood of x point of ,b d 1 II x-x II 0 0 = e. n d = 8 or II x-x0 II d = 0 this is The Taylor expansion for f(x) shows that f(x) = 8 in f nf(x ;x-x ) 0 n For x - x S , say N(x0 ) • contained in not in N(x ) . Since 0 0 E Let y be any is a connected open set it is here also arc-wise connected and the points x 0 and y may be joined by a finite chain of neighbourhoods N(x ) , N(x..), ••• ,,N(x ),, 0 such that y is in N(x ) and N(x.) contains x. n l+1 J. n I • Since x,, lies in I N(x ) , a neighbourhood N' (x,,) C N(x ) can be found in which f(x) 0 I vanishes. 0 The prece¢ding shows that f (x) vanishes in N(4) and hence by induction f ( y ) = e,, that is, f(x) Theorem 3. 7. 2. = e for all x in lJ • If f(x) on L to L' is analytic at all points of 1 and if its values lie in a bounded set in 1 1 then f(x) is constant in L that is f(x) = y 0 , where y 0 is a fixed element of L'. -106- Proof. Let x 0 and x and consider the function f(x S • 0 1 (~- x 0 )) of the complex variable + This is an entire function of } whose values lie in a bounded set, and hence by Theorem 2.6.4 takes on a fixed value, say y t 0 S • In :r:nrticular for S = o, f(x 0 ) =y 0 and for for all Theorem 3.7.3. the domain Let f(x) be a function on L to L • Then if for any d U f(x) ll < M(d ) in 1 in D', sup llf(x) II , x in d3 d ' d' Proof. J = 1, is the boundary t = M(d ), either t II f(x) II , - or else in L , ' analytic in JJ u tf:3 , where 8 and continuous in J1 of 4, be an arbitrary pair of points in 1 M(d ) in .$ d Suppose x is in 0 JJ II f(x ) II and 0 ct' ~ M(d' ). • Let y in L be fixed but arbitrary and consider the linear manifold S I = x + ~ Y 1• S n o5 ~ [ x in L x topology induced in S by L and since x to an open set D. t:l Let 0 D. ~ , i.e., to JJ - D , S n 03 /J. denoting the closure corresponds 11 must contain } = o. be the (connected) component of 4 containing Then ( ;; 0 and y are fixed it corresponds and hence Now x • 0 in the romplex plane. b. to the boundary of of is open in the relative 0 0 is in S n t::. ) C ( 'A - J- = O. The following assertions are then 4 )• valid: (i) f(x + l y) is an analytic function of (ii) f(x 0 0 + } y) is continuous in t::. o' (iii) sup II f(x + ). .) in -A 0 - t::. 0 0 1 y) II , d ' M(d ) • in 11 o' -107- By Theorem 2.5.11, II f(x + ~ y) II 0 d I ~ 1 M(d ) for I II f (x ) II = M( d ) • However, the same theoren asserts that d1 II f(x + ~ y) II 1 = M(d 1 ) everywhere in .1 • But y was arbitrary so Hence 0 d we may take for y any point x in lJ a straight line segment lying in /) 0 0 V'hich maybe joined with x • amounts to saying that the assumption II f (x0 ) II 1 by The above argument then II f(x ) JI 0 a contradiction while 0 = M( d 1 ) I d 1 implies d for every x having the property just stated. > M( d ) leads to II f (x) II 1 =M( d 1 ) d This in turn implies l\r(x)l\ ,=M(d )foreveryxin ~ vlhich may be joined with x 0 d 1 by a polygonal line lying entirely in ~ • Hence II f(x) II 1 :: :M(d ) d if equality holds at a single point of ob • 1 Theorem 3. 7.3. 0 ~ r f(x) 1 < II x II d < r Let f(x) on L to L 2 , d in D. 1 be analytic in the doma:in .& Then f(x) may be expanded in the form Cr> 1 = 2=' pn(x), where pn(x) = 2 Tri /, r f(~x) 1n+l d~ , n = o,_::- 1,,:t 2, ••• , is a circle of radius d < r 2 and r 1 < II x II ,.. r, <f < - 2 about the origin. The fun:! tions pn (x) are analytic ll x lld II x lld f , where r when r < ll x ll d 1 < r 2, and pn( ~ x) = J- np n (x) when } x and x both satisfy this inequality. Proof. For a fixed x in ,/)- , f( }" x) is analytic in S when ClO r, < II X J1d I SI< r _2 JIXlld • Hence by Theorem 2.6.1, f( with p (x) as given above. n '.r x) =I_ pn(x) j n Since the region of convergence of this -108- series includes the point } = 1, setting } =1 gives the desired rerult. § 3. The Generalized Cauchy-PJ.emann Equations 3. 8 The Generalized Cauchy-Riemann Ecµations ( 21 ) • Let L(C) ' and L (C) be the associated couple spaces corre::ponding to the t sequentially complete locally convex real 1. t. s. L(R) and L (R) respectively. D will denote the associated directed system for L(R) ' ' and L(C) vilh.ile D will denote the associated directed system for L (R) f and L (C). f(z) ' f ' n A function f(z) on L(C) to 1 (~) can be written as =r 1 (x,y) + if 2(x,y) where r 1 (x,y) and f 2(x,y) are functions of t two variables on L(R) to 1 (R). I of L(C) to L (C). Let f(z) be defined in a domain The analyticity of f(z) can be discussed in terms of the properties of f (x,y) and f (x,y). 1 Theorem 3.8.1. 2 In order that f(z) be analytic in /) it is necessary and sufficient that the functions f (x,y), f (x,y) be 1 2 continuous jointly in (x,y) and admit at all points of fJ first partial G-differentiaJs which are jointly continuous in (x,y) and that the equations axf1 (x,y; 6 x) = 3.2 f yf1 (x,y; Ll x) = - a x f 2<x,y; are satisfied in r() 6 x) for an arbitrary increment .ll X in L(R). partial G-diff erentials are in fact partial F-diff erentials. The -109- o!J If f(z) is analytic in Proof. it is continuous there, 11 z and continuous and the differential df(z; fl z) is linear in (separately) in z and The contirmity of f (x,y) and r (x,y) 2 1 ti z. is then an immediate oonsequence of that of f(z). df ( z; 6 z) = lim _r__ ( z_+_J_D._z..;...)_-_f,.,_(z_...) • j- _,. O_ Hence in particular taking Now, fl z = j- Llx + i•e, } = t, where t is real, df( z; .6 x) = lim f (x + t Lix,y)-f (x,y) + if (x + t 6x,y)-if (x,y) 1 1 t _,. 0 2 2 t This limit will e.x:ist only if the separate parts have limits. Hence, = ~xf1 (x,y; D.X) + i b xf 2(x,y; b.. x). 3.4 df(z; 6x) Similarly, df(z; ~ x) = $yf 2(x,y; tJ. x) - i ~ y~ 1 (x,y; ~ x). 3.5 Since the left number of each of these equations is continuous in z the G-differentials in the right members are continuous jointly in (x,y), for fixed t6. x, when x + iy is in ~ • For fixed (x,y) in J) the G-diff erentials in the right members are continuous in .6 x. Equations 3.2 and 3.3 follow by equating real and imaginary parts of the two expressions 3. 4 and 3 •5. The partial G-differentials in 3.2 and 3.3 are in fact partial Fdifferentials. Thus, -110- while, df(z;a .6 x +b 6. ~) 1 = a [ S Jtr1 (x,y; £l = adf ( z; ~ 1 x) +bdf( z; l'.'1 2x) 1x) + i ~ xf lx,y; !l 1x) ] + b [ ~ xf1 (x,y; 6. 2x) + i ~ xf 2 (x,y; 11 2x) 1 • I?rom these relations, by equating real and imaginary parts of the expressions on the right hand side, it f ollows that ~ xf (x,y;a 6. x + b A ~) =a ~ xf (x,y; ~ 2 1 2 1 x) + b ~ xr 2 (x,y; l:! ~). If z ::: x + iy, then II x ii d ~ II z II d• Hence II r 1 (x + 6. x,y)-f1 (x,y)- Sxr1 (x,y; ,6 x) II d' ~ ll f ( z+ Ll x)-f ( z )-df( z; 6. x) JI ,, d for d ' in D 1 • '7 > d in D exists so that given such that for lh~ x II d < t O, there is a ' f , II f(z + 6 x) -f(z)-df(z; ~ x) II , d Consequently, for ll~ x /I d < E , II f1 (x+ A x,y)-f1 (x,y)- r xf1 (x,y; A x) /I ~ { /l.D. x ll d, II r 2(x+ t::. x,y)-f 2 (x,y)- '6xf 2 (x,y; c1 x) II d, ~ { ' llll x ll d d 1 and the partial G-differentials are indeed partial F-diff erentials. To prove the sufficiency of the conditions suppose that > 0 L\ z = 11 x + i is an arbitrary element of L(C) and consider the !::!. y expression: f f(z + }- ~ z)-f(z) s 1 (x + s L\ x-t A y, y + t {). x + s Ay) -f1 (x,y) s + i t f +i 2 (x+s A x-t Ay, y + t Ll x + s L1y) -f (x,y) 2 ~~~~~~~~~~~~~~~--~~~ s +i t where z is in of) and } =s + it is a sufficiently small complex nunber. The function F1 (s_, t,u,v) = f 1 (x + s /j x-t JJy,y + u D x + v L\ y) t of four real variables with values in L ( R) is continuous and admits continuous first partial derivatives near (o,o,o,o) (Theorem 3.2.2 and the hypotheses of the theorem.) For a function with these properties, the following mean-value theorem holds, namely, Ol F1 F1 (s,t,u,v) - F1 (o,o,o,o) - s ~ s F1 (o, o,o,o) - t ~ (o,o,o,o) dt & F1 3. 6 'd F1 -u (o,o,o,o) - v (o,o,o,o) = 5(s,t,u,v) 'du d v fl~ (s,t,u,v)ff d' ' ' lim =0 for all d in D • where (s,t,u,v).+(o,o,o,o) I sl + \ t i + tu! + tv I Writing S(s,t,u,v) = (F1 (s,t,u,v) - F1 (s,t,u,O) - v &d vF1 (s,t,u,O)) 'd F1 + (F1 (s,t,u,O)-F1 (o,t,u,O)-s ~ (o,t,u,O)) + (F1 (o,t,u,O) d s ~~ ~~ - F1 (o,o,u,o)-t -(o,o,u,o))+(F1 (o, o,u,o)-F1 (o,o,o,o)-u -(o,o,o,o))+ d t 0 u -112- a F1 a F1 dV 6 F1 -;} V +v(~(s,t,u,o)- ~ (o,o,o,o)) d F1 + t(~ (o,o,u,o)- ~ (o,o,o,o)) . ~ s d t The result now follows on making use of the triangular property of the pseudo-norms, the joint continuity of the partial derivatives near (o,o,o,o) and the properties of integration for functions of a real variable with values in a locally convex l.t.s .(12,. 18). This last is used to write ~~ F (s ,t,u,v)-F (s ,t,u,O )-v ~(s,t,u,O)= 1 1 3 v v j a~ ~~ ~v dV ( ~(s,t,u,v)- ~(s,t,u,O))dv. 0 Consequently, when 3.6 is expressed in terms of partial G-clifferentials, it reads F1 (x + s 4 x-t 6 y, y + t Ax + s A y)-F1 (x,y) = s f xf1 (x,y; D. x) 5 (s, t, t,s). There is a similar relation involving f (x,y) and a quantity 7(s, t,t, s) 2 -t ~ xf1 (x,y; 6 y) + t ~ yf1 (x,y; + s er yf1 (x,y; Cly) + .6 x) such that for any d ' in D' ll~(s ,t,t,s) lld , lim (s,t) -+ (o,o) Isl + ltl + l t l + 1sf 0 • Using equations (1) and (2) f(z + ~A z)-f(z) ~ + i• ( Now l'SI ~ <:' <I [ c( ci xf1 (x,y; A x) - $; xf (x,y; 2 xf 1 ( x, y; tJ. Y ) + ('oxl"' 2( x , y; u x) )] ( S +S it) + J_ ( Isl + f2" A l t l ) and for any d ' in D,' A y)) ~ S"+ i 7 • -113- II ~ + i 1 ll ~ II ~ II + ~ .;r:::;2: : ( Hence, dt lim II ~ + i ~ II d' ~ -+0 l~I II SII d' + . II Yf II d') = o. lim (s,t)+( o,o) I sl + lt l (s,t).+(o,o) /sl + /t i lim Thus f(z) has the differential Since r (x,y) and f (x,y) are continuous so is f(z) and f(z) is analytic 2 1 in J3 • Theorem 3.8.2. If f 1 (x,y) and f 2(x,y) are two functions on L(R) ' to L' (R) where L (R) is complete, and f(z) = r (x,y) + if (x,y) is 2 1 analytic in the domain of the couple space L( C), then r 1 and f of) admit total F-clifferentials in JJ 2 (considered as a domain in the composite space L(K)) and df(z; 6 z) = df1 (x,y;( t:.. x, 11 y) + idf (x,y;( L1 x, !), y)) 3.8 2 6 z = Ax + i where F-diff erentials of r in o8 • 1 A y, and f 2 ti x and D. y being in L( R). The partial of all orders exist and are continuous They have certain symmetry properties (see below) in particular, the generalized or abstract Laplace equation satisfied by the real and imaginary parts of f ( z), 2 2 dxxfk(x,y;( ~ x; 11 y)) + dyyfk(x,y;( fl x; L\ y)) k =0 3.9 d~fk(x,y;( L\ x; D. y)) - d~fk(x,y;( D. x; D. y)) = 0 3.10 =1,2. Proof. Using J.2 and 3.3, 3.7 may be written as 3.11 df(z; 6. z)= $>'x f1 (x,y; L\ X)+ b' yr1 (x,y; A y)+ i( ~ xf 2 (x,y; f). x)+ Syf 2 (x,y; L:i y). -114- For 3.8 to hold it must be shown t!Bt = df1 (x,y; ( A x, A y)) 3.1 2 S' xf (x,y; fl y ) + 'i yf (x,y; tl y) = df ( x,y; ( Ll x, .I),. y)). 3. 13 ~ xf (x,y; fl x) + ~yf ( x,y; 6 y) 1 1 and, 2 2 2 The partial G-differentials occuring in 3.12 and 3.13 are, by Theorem 3.8.1, partial F-differentials. To prove the contention 3.12 df (x,y;( A x, A y)) must be shown to be linear in the increment 1 Li zx., b ':?') ) = df1 (x,y; (L\ 1x +Ar, L\ 1y + tJ i')) = ~ xf1 (x,y; ll 1x + A zx.) + ~ yf 1 (x,y; A 1y + /j ':?f) 1 df (x,y;( ~ x, Ll y)+( 1 1 = S xr1 (x,y; i:J 1x) + ~ Yf1 (x,y; A 1y) + cS' xf1(x,y; Ll r) + ~ /1 (x,y; Ll i') = df1 (x,y;( fl 1x, ll 1y) ) + df 2(x,y;( fl 2x, fj if ) ). To verify that df (x,y; ( A x, A y) ) is a continuous function of the 1 = (8,8). increment it is sufficient to verify continuity at ( Li x, .1 y) Thus given d ' in D ' , II !), x /I d < and ~ , > E '>/ > 0 such that II A y II d < ~ implies II ~xf 1 (x,y; A x) JI d' < ~ II S r 1 (x,y; A y) II y for 0, there exists d in D, f. < - • d' II ( A x, A y) II d = 2 Consequently l/ df1 (x,y; ( A x, 6 y)) II II x 11 d + II y /I d <1 . Further given I 7 > 0 d' in D' there corresponds d in D with the property that given ~ there exists > 0 such that ll.6 z U d < ~ ~ II f( z + A z)-f(z)-df(z; A z) II t and recalling that if x ~II x dI 1 implies Hence i f II ( A x, A y) "d < I I + iy is an element of L (C) then II x lid, ' + iy ' II ,, it follows that d 1' llaz II d. < ~ di r, -115'- II £1 (x + A x,y + /1 y) - r 1 (x,y)-df1 (x,y;( /1 x, A y)) II d' ~ II f(z + ~ z)-f(z)-d.f(z; D. z) /I d' ~ 'Y/ ' II ~ z II d ~ 1 ' ( 11 fl x II d+ 116. y 11 d) = '? II ( L\ x, .6 y) lld. 1 Thus df (x,y; ( A x, /!. y)) is in fact the total F-differential of r (x,y). 1 1 Similarly df (x,y;( A x, f1 y)) is the total F-differential of r (x,y). 2 2 Vfuen f(z) is analytic it has F-d.ifferentials of all orders which are jointly continuous in the increments and symmetric IIUlltilinear forms in these increments. f 2 From this the further properties of r 1 and stated in the theorem follow. Thus when Ll z = ( Ll x, f,, y) , d.f(z; ~ x) and d.f(z; A y) are differentiable functions of z and the expression 3.4 applied to them leads to 2 d f(z; d x; b y) = d~f 1 (x,y; D x; f). y)+ id~f 2 (x,y; LI x; /J. y) J.14 and 2 d f(z; /1 y; A x) = d~J/ 1 (x,y; /1 x; b. y)+ id~/ 2 (x,y; L\ x; b. y). 3.15 2 But d f(z; d x; Dy) = ct 2f(z; ~ y; /1 x) and on comparing 3.14 and J.15 2 and 2 d:xxf1 (x,y; /j x; /1 y) = dxxf1 (x,y; A y; L\ x) 3.16 2 2 dxxf 2(x,y; ~ x; !J y) = d:xxf 2 (x,y; 3.17 f). y; ~ x). Similarly using the equation 3. 5, the expressions 2 c· ) 2 c ) dyyf 1 x,y; /1 x; /j y = dyyf1 x,y; /1 y; /j. x 2 dyyf 2(x,y; /1 x; A y) and 2 = dyyf 2(x,y; A y; ~ x). are valid. By considering 3.4 as the given analytic function and writing 3.18 3.19 -116- down for it the form of the differential given by J.5 it is seen that 2 d f( z; Ll x; D. y) = d~f 2(x,y; {). x; /J. y)- i d~f1 (x,y; /J x; A y) 3. 20 On the other hand the expression J.5 may be considered as the given analytic function whose differential is given according to the expression J.4 by Then comparing 3. 20 and 3. 21, the expressions of 3. 10 are obtained in 2 and 2 d.x;/1 (.x,y; tJ. x; (j y) = dyxf1 (x,y; L1 x; Ll y) J. 22 2 2 dxyf 2 (x,y; /j x; Ay) = dyxf Cx,y; ~ x; /J y) 2 J. 23 Other relations, of the form dxyf1 (x,y; ~ x; ~ y) 2 2 =dxyf 1 (x,y; 6 y; ~ x), J.24 2 2 = dxyf (x,y; /j y; !J. x), 2 J.25 dxyf f(x,y; ~ x; (j y) 2 3.26 and can be obtained similarly. Taking the partial F-differential with respect to x, with increment fj y, of both sides of J. 2 gives and from 3.3 on using J. 23 2 dy:/1 (x,y; D. x; ~ y) 2 = -dxyf 2(x,y; A x; L\ y) 2 = -dyxf 2(x,y; L\ x; A y) 3. 29 -117- Comparing J. 28 and J. 29 gi. ves Similarly In concluding this thesis some colJUllents of a historical nature are appropriate. 'rhe earliest work on a theory of analytic functions in normed linear spaces was done during the year 1931-193 2 by Professor A. D. Michal in collaboration with R. s. Martin. In a statement in ( 27) Professor Michal himself supplies details on this work which was done in his sa~inar at the California Institute of Technology. These same men continued with this study and in 1934 introduced the concept of a normed linear ring (28). With regard to couple spaces the pseudo-norms defined in 'l'heorem 1.5.1 are analogous to certain norms used in couple Banach spaces. Michal and Wyman, in 1941 ( 29), gave another definition of a norm for couple Banach spaces. The Cauchy-Riemann theory considered in Chapter III is a generalization of work done by Taylor ( 21). Further work along this line was done by Michal, Dav:Ls and Vfyman (JO), who gave an introduction to a theory of polygenic functions in normed complex couple spaces. -1'18- REFERENCES (1) s. Banach, Theorie des operations lineaires, Warszawa (1932). (2) E. Hille, Functional analysis and semi-groups, New York (1948). (3) J. Dieudonne, La duali te dans les espaces vectoriels topologique, Annales de 1 1 Ecole Normale Superieure, vol. 59 (1942), pp. 107- 139. (4) J. von Neumann, On complete topological spaces, Trans. A.M.s., vol. 37 (1935), pp. 1-20. (5) A. Tychonoff, Ein Fixpunktsatz, Math. Annalen, vol. III (1935), pp. 767-776. (6) J. N. Wehausen, Transformations in linear topological spaces, Duke Math. Journal, vol. 4 (1938), pp. 157-169. (7) A. Kolmogoroff, Zur Normierbarkeit eines allgemeinen topologischen linearen Raurnes, Studia Math., vol. 5 (1934), pp. 29-33. (8) D. H. Hyers, Pseudo-normed linear spaces and abelian groups, Duke Math. Journal, vol. 5 (1939), pp. 628-634. (9) S. Mazur and w. Orlicz, Sur les espaces metriques lineaires, I, Studia Math., vol. 10 (1948), pp. 185-208. (10) J. Dieudonne and L. Schwarz, La dualite dans les espaces ( ':F ) et( ..tJ; ), Annales de l'Institut Fourier, Univ. de Grenoble, vol. 1 (1950) pp. 61-101. (11 ) G. Birkhoff, A note on topological groups, Comp. Math., vol. 3 (1936), pp. 427-430. (12) A. D. Michal and E. W. Paxson, The differential in abstract linear spaces vci.th a topology, Comptes Rend. de la Soc. des -119- Sciences de Vorsovie, vol. 29 (1936), pp. 106-121. (13) A. D. Michal and E. w. Paxson, La differentialle d.ans les espaces abstraites lineaires avec une topologie, c. R. Acad. Sci. Paris, vol. 202 (1936), pp. 1741-1743. (14) R. Arens, The space 1 and convex topological rings, B.A. M.S., vol. 52 (1946), pp. 931-935. (15) J. P. La Salle, Pseudo-normed linear spaces, Duke Math. Journal, vol. 8 (1941), pp. 131-135. (16) H.F. Bohnenblust and A. Sobczyk, Extensions of functionals on complex linear spaces, B.A.M.s., vol. ~ (1938), pp. 91-93. (17) G. W. Mackey, On convex topological linear spaces, Trans. A. M.S., vol. 60 (1946), pp. 520-537. (18) E. W. Paxson, Analysis in linear topological spaces, Cal. Tech. Thesis (1937). (19) v. Ganapathy Iyer, On the space of integral functions, I, Journal of the Indian Math. Soc., vol. 12 (1948) PP• 13-30. (20) E. c. Titchmarsh, The theory of functions, Oxford, 1939. ( 21 ) A. E. Taylor, Analytic functions in general analysis, Cal. Tech. Thesis ( 1936). (22) A. D. Michal, Differential calculus in linear topological spaces, Proc . N. A.S., vol. 24 (1938), pp. 340-342. (23) A. D. Michal, Differentials of functions with arguments and values in topological abelian groups, Proc. N.A.s., vol. 26 (1940) 1 pp. 356-359. (24) A. D. Michal, Functional analysis in topological group spaces, -120- Math. Mag., vol. 21 (1947-48), pp. 80-90. (25) D. H. Hyers, A generalization of Frechet's differential, Revista de Ciencas, vol. 49 (1945), pp. 645-663. (26) R. s. Martin, Contributions to the theory of funct.ionals, Cal. Tech. Thesis (193 2). (27) A. D. Michal, On a non-linear total differential equation in normed linear spaces, Acta Math. Vol. 89 (1948), pp. 1-21. (28) A. D. Michal and R. s. Martin, Some expansions in vector space, Journ. de Math. Pure et Appliquees, Vol. 13 (1934), pp. 69-91. (29) A. D. Michal and M. Wyman, Characterization of complex couple spaces, Annals of Math. Vol. 42 (1941 ), pp. 247-2.50. (JO) A. D. Michal, R. Davis and M. Wyman, Polygenic functions in general analysis, Annali de Scuola Normale Superiore de Pisa, Vol. 9 (1940), pp. 97-107.