Contributions to the Theory of Functionals Citation Martin, Robert Samuel (1932) Contributions to the Theory of Functionals. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/9rtv-x659. https://resolver.caltech.edu/CaltechTHESIS:12222021-155415955 Abstract Part 1. Polynomials and analytics in vector space Part. 2. Various theorems on the representation of functional forms and transformations by means of Stieltjes integrals Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Mathematics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Michal, Aristotle D. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1932 Record Number: CaltechTHESIS:12222021-155415955 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:12222021-155415955 DOI: 10.7907/9rtv-x659 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 14459 Collection: CaltechTHESIS Deposited By: Kristofer Jolley Deposited On: 22 Dec 2021 18:23 Last Modified: 09 Feb 2023 20:36 Thesis Files PDF - Final Version See Usage Policy. 38MB Repository Staff Only: item control page

CONTRIBUTIONS TO THE THEORY OF :BUNCTIO.N.A.LS Thesis by Robert S·. Martin In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFOfilUA INSTITUTE OF TECHNOLOGY Pasadena, California 1932 Part 1. - POLYNOMIALS ~ TD ANALYTIC ::B'UNCTIONS IN VECTOR ·SPA.CES 11. Introduction to Part 1. Part 1 consists of three divisions. In division (12) vector spaces are defined and some of their immediate properties de duced. Division (13) discusses polynomials and deduces a number of their pro' ¥ ~rites. In the final division the work of the preceii ng pages is ap_plied to a gener al ization of the ordinary t~eory of analytic functions. 12. Vector Spaces 12.l General RGmarks. Definitions have be ~n given by Banach, Weiner, Frechet, and others, of abstract spaces naving properties similar to those of the space of ordinary vectors. In order that this paper may be self contained and in order that there may be no c·onfusion as to what is meant, we shall give our own formal statement of the postulates defining Vector Spaces. Our choice of t he particular system has been guided by a desire to he.ve, • later on; a s many analogies as possible to the ordinary theory of functions. 2, 12.2 Vector Spaces and the N0 tation The spaces with which we deal are systems consisting of a set II ..• jj . E of elements or points, a number system A, and three operations $, ~, T'nese systems are as sumed to satisfy so r-ie or all of the postclates given below . The set Eis called the su~port of the space. E are usually denoted by the l _e tters x, y, s,... . The elements of In general no ambiguity aries when we refer to the space and its support by the same letter. To emphasize the fact that a space E involves a particular number system .11 , it will be referred to as an E(A1 ). Functions, Mapuing : The usual notion of mapping is assumed. If to each el ement x of a set E0 there corresnonds by some law a well determined x 1 of a second set E;, the set E 0 set E~. A function f is a syrribol for such a mapping. denoted by x1 = f( x). to f is said to :,e map-ped upon tr.e The relation is Following the phrasing of E. H. !foore , we refer as a function on E0 to E~. Corr.1'.)osition of Classes: If Ei_,E2 , ••. ,~ are sets of el ements , n then by E1 E2 •• ·En or ~Ei we shall understand the nrultiplication class 1 of ordered sets (s1,xz,•·•,xn), where x 1 runs through the set Ei. n E1 = E2 =•·•=En, ~Ei is denoted simply by En. The set If E1 is spoken of as the cornposi te of tre sets E1 , •••,En. Functions Involving Parameters: If f is a function on E0 to E10 n where E0 = 7TE1, it is sometimes convenient to consider the properties i=l of the sub-mapping obtained by h olding fixed, certain of the el ements x 1 , A qua 3. ~unction of ~,··•,xic· In this E1 - •• ~(Ek+i•··EJ to E~. f will be said to be a function on The elements s 1 ,•••,xk of E1 , ••• ,~ will then be spoken of as variables; the elements xk+i,···,xn of Eir+i,·•·En as param et er s. 12.3 The Postulates for Spaces E(R) and E(C ). A1 : ~ is a class consisting of at least two elements. A2 : A is either R, the real number system, or C, the complex number system. B1 : z = riy is a function on E2 to E. That is, to each ordered pair x,y of elements from E there corresponds a unique $ 1..4.M element z called their~. 132: x-ey = yfx %: (xey)ez = ri(~z) 134 : z = c:,<_ ~x is a function on AE to E. That is, to each ordered pair (q ,x) consisting of an element ~ of A and an element x of E there corresponds an unique element z of E called the product of o· and .x. %= The set Eis included in the set (~~k). That is, to each element z of E there corresponds at least one pair (~, x) of AE such that z = (j_Qx. Bs: C(~ (xey) = C( ®x ~ d9y. ~ -: (CX + ) 9 X = o{ f> X ~ .. G x. Where 11 + 11 denotes ordinary addition in the number system A. 13s: (Cl{• ~ )Gx = c< Q( i-: Gx) in A. where"•" denotes ordinary nmltipHcation p = II x 11 is a function on E to the non-negative real numbers; :B : 9 that is, to each x of ~ corresponds a non-negative number P = !j x/1 · \1 x \\ is called "norm of x 11 • B10 : There is a unique x• of E whose norm is zero. Bn: ii()( ~X 1 = ! ('{ , • '. x 1 1 where/ .. ,! denotes the ordi nary mod-..:.1.us 1~., :1 ~ ·! • , I Dl2: : A.~., /• - q X :· + II y I • '!) Notation: \'/hen no ambiguity arises ♦ will be written simply+ and 0 will be omitted entirely. x - y will be defined as x + (-l)y. x• which by B1o and B11 is s een t ~ be identical with O•x for every x will be denoted by O. neighborhood: rrh e t.' 0 int S X for Whi ch '. '. X - the D neighborhood of x 0 • Region: are said to lie in The ~ neighborhood of x 0 is denoted by (x0 )p . A set of points E0 is a region in E if each x 0 of E0 has a p neighborhood (x 0 )P whose points are all in E 0 Connected Region: A region E0 ·is said to be connected if when x x 1 are any two of its points there exists a cnain of neighborhoods finite in number, say (x)p , (x 1 )e ,••·,(Xo)p ) (s1 )p1, each having a point i ~ cormnon 1 0 with the ~receding. Limit Point: A set of points E0 of E is said to have a limit point x 0 if it con tai ns points X disti nct from x 0 in every (x0 )p . An infinite sequence (x1 ,x2,···~,···) of points of f is said to have a limit if there is a point x of E such that to each p there corresponds an integer Np such that for n > Np, xn belong s to ( x )p. sequence. x is called the limit of the c1 : The necessary and sufficient condition the t a sequence(~) have a limit is that for each p there is an l( f:> such that for n :>Np and p >O, ¾-p belongs to (~)p . S~aces which satisfy t he po stulate c1 are said to be complete. r-11 an :/ of the theorems wn ich follow are proved without the postulate of completeness. When co~ple t eness is employed it will be noted. 12.4 Imrr.ediate co nsequences of the Postulates Tl. l·x=x Proof; There exi st s Q~Y such that x = ~ -y (B5 ) Then l • x = l · (r_i • Y) = ( l •q ) · y = Cl 9y (:Ba) X = 19X T 2. O•x = x• Proof: II O·x / = /o/j/ x/ = 0 Hence T 3. X O·x = x• + x• = X Proof: x+ x* = l·x + O•x = (1 + O)x = l·x = x :ByLJ,~ B7 T 4. X - X= Proof: T 5. x• X - X = x+ (- l) x=l·x+ (-l)x = (1-l)x = O·x = x• The s et E forms an Me1.ian group under e. Proof: ? rom B1 , B2 , B3 , follow closure under~, associativity, and commutativity. From T 3 follows the existence of a unit; from T 4 the existence of an inverse. (. horn T 5 the next thre e standard theorems on A~elian groups may be ta.ken over at once. Thus ~_§_ The equation x + z = y ha s the unique solution z = y - x. T 7 The inve r se to xis ~ique, i.e., for each x t here is a unique x 0 (viz. -x) such that x + x0 = x*. T 8 The unit is unique; i.e., there is but one x 0 (viz x*) such that x + x 0 = x. T 9 I f E(A) is c omp le t e and C-Xn,) is a sequence such that n II xn+l - xnll converges then ( xn) has a limit. f Proo f : Let p > 0 be arbitrary. Then choose N so tha t for n >N '.l ~ l - xn l! + · · • + . xn+p - xn+p-1 ') <. (J ( al 1 p > 0). Thus x_ + II X n+p- X n II ~ - iii! X n+l--n Hence by c1 there exists x such that lim ¾ = x. n+oo T 1 :) If a sequence (JSi) bas a limit x then thci t limit is unique. Proof: Let x1 be ano ther li mit Tr1en ii x 1 -x !l = p > 0 Choose N such that for n > N and II~ - x 1 Jl ~ P/2 then /j x 1 - T 11 : Xn - x ;; .:::.. fJ/2 xi/< {l If a sequence (xn) has a limit x, then ll xn ll has a limit and . 7 I lim Xn l = n -roo lim II Xn il n-+ oo Let 1:-:' ->0 be arbitrary. Proof: for n :>ti I Xn - xj/ L F' Choose W such that I T'nen c-0 >/Ix.__ x ~,·1 , -n - x/',I ~ Il ii --n xn j = ;: x ;, = lim !I Xn Ii n +oo Y ,, I ~ 1, X ± y II 11 :,· , I ; X ± y' , Similarly ·. Y ,l - Hance T 13 • j Hence 11 lim n -,. oo T 12 ·111 X /11 - 1 ··x:. ,1 ·, I l;x ~ ! x ± y/i 1 x 1,· - , , Y ,, I s- l'l x ! y I: If (~ n) is a sequence of n11r.1bers of A having ~ as its limit and x &s an element of E(A) then the sequence (~ nx) has as its lblit o; x. ?roof: Choose .~=-- o a r bitrary. Choose N such tha t for n >N jo<.n - '="' j < Then for h > N, ;, <:rnx - ~ x 1,I' = ,I•:)' n - T 14 If · x - Y It', = o, then X= y I Pr oof: T 15 X - y = x• X = x• + y = y + x• = y If qx = x• then either Proof: ex = 0 or X = x• I~ I Jx :.· = ,1I' ~ x /1/' = ,,· x• = 0 Hence either :~ ' = 0 or :i x'. 1 = 0 Hence ex = o or X = x• 01 1 / / fi x'; 1' :1x 1'i' T 16 If ex x = {}X and x :/ x•, then ~ = 8 Proof: C\ X - $ X = x• - -3 = 0 , c1 12 a re all s atisfied if Eis taken to be A itself,~, and 0 i~te~reted as Consistency of the Postulates: The 9ostulates A.1, A , B1 , ••• ,B 2 ordinary addition and rrmltiplication and // •• . If interpreted a s ordinary modulus. R and Q ~ Vector Snaces: With the abo~,e interpretations of ~, 0, II ••• II , the nur:1ber syst ems Rand C may themselves be regarded a s sup;JOrts of vector spaces. There are, of course, other def i~itions of the three o~erations by wh ich B. and C may be taken a s SU3J0Orts; however, for our pur -:oses this 0 d e f i ~it i on is the si ~plest . Function S-pa.ce ~~Vector Space: If the elerr.ents x of E are taken to be re al continuous functions of one or fini~el~ more real variables defined over some fixed doma in D of the variables. If the sun of two functions is defined in t he ordinary manner, and similarly the p roduct of a function msx /xl, t h en it is easily by a constant, and if /j x /j be defined as t he verifie d that with suitable restri~tions upon D t he set of elements x constitutes a vector space E(R). In pa.rtic .1lar if Dis taken to be the 1 linear i Jt erval (a,b) and xis a function x(t) continuous fort i n (a,b), t h_en with the def ini tion I x 1'I = rpax : x( t) j the s·o ace E(R) of elements x \a,b) is the well known space of continuous func t ions. It will in genera l be 1 ... denoted by F(a,b). Couroosition of Vector Soaces T 17 Let E1 (A),•·•,~ (A) be vector spaces ~it r1 the number .ft E be the. composite of E1 , ··•,En their respective system A. Let E == sup _orts. Let the sum of two elements (zt,x , • ••,xn)' (x ,x l=l 1 1 2, · ·•,~) 2 of Ebe defined by Let the product of ~ of A and (x 1 ,x ,•••,xn) of E be defined by 2 (2) Let the norm of (x1 , · · · ,~) be defined by Then the s ys tem c onsisting of E, .A., e, e, • • • I: satisfies the postulates ·A1 , · • ·, c and is therefore a vector space. 1 ~e shf ll verify the post ~lates for t ~is system in order. Ai= The multi~licative class of n classes each having at least 2 el em ents has not less than 2n el ements. A : 2 Tri vial. E1 : ( E : 2 Clear. E : 3 Clear. B4 : ( l'2t) uniquely detem i nes the :product. B5 : 10(x1 ,x , ••• ,~) = (10x 1 ,l~x2 ,•••,l~xn) == (x 1 ,x2 ,•••,~) 2 B6 : Clear . ~: Clear. B8 : Clear. ll)l ) uniquely detemi nes t he surn. I <::J :B9 : (3) defines p = // (x1 , • • • ,~) ~ on E to non-negative real numbers. E1 o: !l (xt ,•·•,x;)II 3 greatest of /i xf j/ ,···, /jxri .l = 0 I Moreover, if greatest of x1 ' · · • ' xn -- 0 then x1 = xi, x2 = x2,··•,Xn = ~:B11: //~ 0(x1 , • • • ,~)// = // (o: x1, • •• A xn) // = greatest of / ~x~~ ··•,I!~ xJI = gr eatest of , ~ // x1 j/ , • •·, !CY, ; i/ Xziii 1 = I ex l312: / I II ( x1 • .. ~) I/ ll (x1 ,•.••,x0 )e(x1,•••,x~) I/ = /l (x1 + x1,···,~ + x~) /1 = greatest of f/ x1 + x1/ , ···,I/ ~ + ~ greatest c1 : Let y. • h of // x1 /j , · • •,/ X_----n /! + greatest of :·I x l i1 ' 1 1 (xh •·· .,!)-) l' ' n rn7Np // y - 1;· • • • ' ,: x' -n,·, !' (h = 1,2, ..• ) be a sequence. First as sune (yx) converges toy. that for ~II ym / < P/2. Choose F and Np so Hence llYm - Ym+p l/ < P If for each p there corresponds Np such that if n:;::,, Np IIYm - Ym+pll~ r then we have II rf - xrP/1 < p . If there- fore E1 ,···,~ are complete each s equence (x1f3) converges to a limit xi. The p oint y• (xi,•··,~) is then the limit of (yk) 12.5 Continuity: Continuity and Continuous Functio~s Let E0 be a region of E. Let f(x) be a func t ion on Eo to E'. f(x) is said to be conti nuous at a point x 0 of E0 if to ~ > 0 determined p > 0 such that if x is in (x 0 ~ then f(x) is in t here is [f(yJ1. If f ( x) is continuous at each point of E0 it is spoken of as conti nuous throughout E0 • ,1. If f(x) is on a region E0 to E' and if ~ >0 there is Unifonn Continuity: detennined p>O such that for ~.?.l' x 0 of E0 , x in (x 0 )p imulies f(x) in [ f(x 0 ~~, then f(x) is said to be u..~iformly continuous in E0 • It is clear f rom the very definition that if f(x) is uniforml y continuous in E0 it is Theorer: ,s .Q.U Continuity Theorem (12.51) If f 1 (x), f 2 (x) are functions on E0 to E' and if (1) the y are continuous at a point _x0 of E0 ~their sum i s continuous at x 0 (2) they are continuous t hroughout E0 ) their sum likewise is continuous throughout E0 • (3) they are u .ifomly continuous in E0 >their sum i s uniforml y continuous Proof: ( 1) Let that x in {x0 ) .:> '1 f 2 (x) in in be given and let f 1 and p2 be so determined ~>0 implies f 1 (x) in [ f(x 0 )] [f2 (x 0 )_, ~ 12 . [f1(x 0 ) + f 2 (x 0 and x in (x 0 ).o implies \2 T"r1 en if ?~ -: 1 , ~ , x in (x0 ) ,.::, i mplies f 1 (x) + f 2 (x) ' 0~for if X E /2 is in (x0 )f) l/f 1 ( x) - f 1 ( x o ) :i c:::. e /2 \\f 2 (x) - f 2 (x0 )11 < E-/2 ;ir 1 (x) + f2(x) - [f1 (x0 ) + f (x0 il// L. € 2 (2) Appl y the argument of (1) to each point x 0 of~- (3) Appl y t he argu~cnt of (1) observing that the choice of p1 and r2 and hence of e does not involve the choice of the particular Corollary: By complete induction a similar theorem holds for the sum of a f i nite number of functions, f 1 (x),··•,fn(x) on E0 to E'. Theorem (12.52) (1) If E0 and .A.0 are regions of E(A) and A res- pectively, f(x) is a function on E0 to E'(A')) g(~) is a function on A~ to A' and if at x 0 of E0 and ~ o of A0 f(x) and g(~) are continuous, then g(~)f,x) is a function on A0 E0 to E1 (A' ) cor.tinuous at (~ 0 ,x 0 ). (2) I f, moreover, f(x), g( Q' ) are u:-.iforr ly c ntinuous and //f(x) /1 , lg ( x ) / are bounde d then g(~ )f(x) is un ifor.:ly continuous. Proof: (1) Let E:>0 be given. rc~',xDP we shall have ( ',x) is in i 0 Choose f\ so that if x is in - ~o l.(f2 (2) Let M be the greater of the upper bounds of jjf(x) jj , jg(~ ) j in B0 and ~o res~ectively. e Let G >O be given and let P1 2 be so chosen )Je- i mnlies f(x) in f (x 0 and e¥ in (~ 0 )p i mplies g(~ ) 11 a~ 2 in {i;(~ 0 )) ~ for all (<f 0 ,x0 ) of A0 E0 . Let ~ = min ( f i f 2). Then for that x in (x0 ) 0 (Cl! , x ) in 2M [(~· 0 ,x0 )](), 1/ g(CX )f(x) - g(C{' 0 )f(x )/I ~ ,' g (t\ ) - g( <r 0 0 )! l f(x) ll + l g(~0 )! ilf(x) - f(Xo) ll 1 l Theorem (12.53) If f(x) is continuous on E1E2 to E' then it is continuous o~ E1 [Ea) to E'. llx - x 'Ii = !j(xl - xl, 0 )II = llx1 - x{II llr( x) :But f( x' ) I! -+- 0 wit h Functions of Functions : 11 x - x'\1 and hence with I x1 - x111 . If E, E 1 , E11 are vector spaces and E0 , E6, Eg are subspaces, pr Jper or not, and if y = f(x) is a function on E0 to E6 and z = g(y) a func t ion on E6 to E~ then z = g [f(x)] is a f unction on 11 E0 to Eo· Theoren: ( 12.54) If EO' E'O' E011 are regi ons and y = f(x) is cont i nuous on E0 to E~ and z = g(y) is continuous on E~ to E~, then z = g(f(x)Jis continuous on E0 to E~. :.: oreover, if y = f( x) and z = g( y) are u niformly continuous in their r8£-pecti ve ret ions) then z = g [f(x)] is u niformly continuous. Proof: Let y 0 Let € > 0 be chosen., and con sider a f,i,xed poi nt x 0 = f(x 0 ); z 0 = g(y0 ). Let f\> 0 be so selected that y in (y 0 \) i n~!)lies z = g(y) in (z 0 )~ . Let p > Obe so selected that x in 2 l (x0 )f i~plies y = f(x) in (y0 )p . Then if xis in (x ) p , z is in 2 (z )€. 0 1 0 2 If f(x),g(y) are unifonnly continuous in their resp ective reg ious, the choice of p a nd in particular of ~ does not involve the point x 0 • 2 1 Corollary: By complete i nduction it follows that a similar theorerr. holds for any finite number of successive functional operations. 14-, 12.6 Other Special Notions Successive Differences: an elerr,ent of E. Let f( x) be a function on E to E' . Let .1 x be Then ~ f(x) = f(x + ~ x) - f(x) is a function on E2 to 1 1 and is cail.led the first difference of f(x) with respect to x. If L1x,•••,Anx denote elements of Ethen the nth difference, (n) ,1. f(x) a t. n-'--n-i··· A1 f(x) of f(x) with respect toD 1x,•·•,L\nx is defined inductively by !!. ntJ.n- i· • • A1f(x) ;; Dn (~ n-1 • • • l::. 1f(x)) =A n- i· •·A l f(x + LJnx) -f\i-l • • • Al f( x) (n) 'Ihe nth difference is written in the form fl f(x) in case there is no ambigu.i ty as to what is meant. Theorem (12.61) Proof: ( n) L1 f(x) is clearly a function on r+1 to E 1 • (n) t1. f(x) i.s syrcrn etrical i n t he increments For any h( x) on E to E' and any pai r~ 1 x and ~x of the increments we have + h ( x) = 6.j '1_h ( X) (n) Hence at any s t age in the expansion of 'S· A f(x) we may exchange two adjacent But any permutation of the .6,1 s may be produced in this manner. '.5. Theorem (12.62) If f1(x),·••,f 8 (x) are functions on E to E', then~) regar~ed as an operator, operates distributively upon the sum f \x) + f (x) + • •• + f\~(x). 1 2 Prooff Let f(x) e f 1 (x) + f 2 (x) + ··· + f (x), then for n = l 8 6. f(x) = L_ fr(x +6x) -Lfr(x) = ~_f/x + ~x) - fr(x) = ~fr(x) Using this result and assuming the theoreD for (n-1). (j)f(x) =An (n_zl)f(x) Theorem (12.63) (n) 6. =~d- (n~l)fr(x) =L (i)fr(x) regarded as an operator commutes with nunerical m1.1.l ti pliers . Proof: Let f(x) be on E to E 1 (A) andO( be in A. Then Thus assuming the result for (n-1) Theorem (12.64) Ifd (A) is a function on A to A1 and a. is s ome fixe d element of E(A' ) then for any increments Al~' .•• , Al re have Proof: Exactly similar to (12.63) except that the multiplier is now on the right. Homogeneous Linear Continuous Functions: In order to make the next section intelli gible we mus t define here a special case of homogeneous CJ polynomials which will be discussed in 13. A function f(x) on E(A) to E'(A) is homogeneous linear co ~tinuous if (1) it is continuous. (2) f(C\' 1~ +cx~2 ) = o< f(x 1 ) +C>C/(x ) for all x 1 ,x 1 2 2 in E and all cx 1 ,~ in .A.. Dif f erentials: Let E0 be a region of E. A f~nction f(x) on E0 to E' is sa id to have a differential at a point x 0 of E0 if there exists a function fx(x 0 ;.t:.x) on E[Eo.J to E' soch that (1) f X (x0 ;.Ax) is conti~uous h01r.ogeneous linear on E(EoJ to E'. (2) E (Ax) defined byE (Ax) = fx( xo;~x) -A f(x) ( ~ x · :/. O; (x0 + lJ x) in E0 ) l~ H E(0)=O is contin~ ous atA x = 0. fx(x ;~x) is called the differential and will be denote :.. in this way or by t he spnbol df(x 0 ), the argument 6 x being 0 , understood. Higher Differentials: If df(x 0 ) a fx(x 0 ;~ 1 x) exists at each x 0 of E then df(x 0 ) is on E~ to E'. 0 The second dif f erential fx2(x 0 ;~1 x,6.2x) is defined as the differential of fx(x 0 ;~ 1 x) q~ function on LE]E to E'. 0 The second differential will also be denoted by d1 d2 f(x 0 ) or in case there is no ambiguity by d2 f(x 0 ). Higher differentials are defined in- ductively in the same manner. Calculation of irut Differential: This is simply the generalization to vector spaces of the theorem due to Gateaux. Theorem (12.65) If f(x) on E0 to E' has a differential df(x 0 ) at x 0 of E0 t hen Proof: By definition we have that Replace Ax by ~.1 x and divide by X fx(x ;l1 x) =- f(x 9 +>-Ax~ - f(x 0 ) 0 I A 8 't-- ➔ O, E- Wx) ➔ O. Hence the theorem. The remaining necessary theorems on differentials we shall postpone until the discussion of the properties of linear functions has been reached. 13. Polynomials ·1n § Vector Space 13.l Introduction Frechet in his 1910 pape~ discussed ptlynomials in the space of continuous functions. In a later paper .. (1928 ; 11e generalized many of the theorems of the previous _pap er to a class of spac es which he calls 11 espaces algebrophiles 11 , and of which our space is an instance. The essential result in both t hese ~apers is that a general polynomial is -;,_ uniquely re~resen tab le as a sum of horr.ogeneous polynomials. The lefiition taken by ~reche t in his later paper, w11ile entirely elegant and satisfactory for spaces closed under multiplication by real numbers--which are the only ones he considers--runs afoul of the common definition of a ~olynomial in a single complex variable, if taken over bodily to spaces closed unter multi~lication by complex numbers. In order that later generalizations of functions of a complex variable may be made, it is essential that we formulate a definition which will be equally valid for spaces E(R) and E(C). ~he definition we have taken is equivalent for spaces E(R) to the definition of Frechet. Half the equivalence is really proved quite incidentally in Frechet's paper. We shall give an independent proof whose details are quit e simple. In the latter section of this work in polynornia1s, we shall discuss modular properties of polynomials and their related forms. We shall point out distinct ions that exist betwe en the modular properties of polynomials on E(R) and of those of polynomials on E(C). Tnis latter part of the work, we hardly need mention, does not presume to be in any way exhaustive of what looks to be a rather large subject. •M. Frechet, Ann de l 1 Eco le Normal Sup. Ser 3. Vol 27, 1910. ••M. Frechet, Joun1cd ~e Mo..-41-\ . ~2~. 13.2 Polynomials QB. A to E(A) Definition: A function ~(A); 0 on A to E(A) is called a polynomial, if it is expressible in the forrr. where the summation extends over a fi ni t2 number of terms, where ~ r stands for the ordinary rth power of)-_, and where each ~ is a fixed el ement of E(A). If n is the highest index for which ar / O, n is called If the f unction p()..) = 0 for all I'- of A, then p(~) the degree of p(~). is ca lled a 11 null 11 polynomial, and we add the g loss that the null poly- nomial may be. regarded as having any degree whatever. Theorem (13.201) If p(,\) is a polynomial on A to E(A) it is continuous. Proof: By Theorems (12.52), (12.51), p(~) is the sum of con- tinuous functions each of wi1 ich is continuous. Theorem (13.202) If p(/') _is a polynOlllial of degree non A to E(A) and if g().) is a. polynomial of degree m on A to A, then g()J:p(f) is a polynomial of degree n+m on A to E(A). Proof: Let 2 c) The first two steps f ollow from the associati~e and distributive laws. ~Jow the coefficient bn+rn is simply Ci/man. But by definition~ J O, an ~ 0 and hence bn+m ~ 0. If either or both p(~) and g(/\) are null, their product is null and. the the orem is still true. Theorem (13.203) If p1(~), ••• ,Pm(~) are polynomials on A to E(A) m each of degree ~ n, then a linear combination p(>--.) • ~ Ci p (/\) 1 1 i=l with numerical coefficients ct i of A is a polynomial of degree ~ n on J. to E(A). Proof: p 1 ().,), even though not of degree exactly n, may be n r1ritten in the form p 1 (t) = null. Y' /\rair by taking certain of air to be r:0 Then n Theorem (13.204} E&ch coefficient ar of a polynomial p(~) ~ ~/'rar r=O of degree non A to E(A) is exnres s ible as a linear .eompination i of the n+l vectors p(O),p(l),··•,p(n), where Ars are numbers indepe.ndent o:~ the choice of the polynomial p()v. Proof: Consider the determinant ( OC sr ( , where C\r = sr J-- ' ' ,, (r,s = 0,1,···,n) a00 = 1. and its value is This is an instance of Van~c.nd 1 s determinant, ,1t (i • j ) /: 0 . i>j Let Ars· be the typical tenn of the inverse determinant, so that r J k r = k . n now p(s) = L n s~ K k=O . Theorem (10 .205) = YC( ka. k:::O s - lC ~•v,o po lynomi als p(") and (l(/\) on A to E(A) are equal for all values of~ if and only if their coefficients are equal. Proof: Then by (1 3 .203) Let the degrees of p°') and Q"/1) be ~ n . . r(A) is a polynomial of degree~ n. Moreover, if Sp, bp , cp are respe ctivel y the coeff i cient s of ~p in p(~), q(~ ), r( ), then a gain by (13.203) By hypo thesis r(A) = 0 for all~. Hence by (13.204) a..... ... b ~.,.. 13.3 er= 0. Therefore p Characterization of Poly nomials £a g to E(R) We shall in this secti on give a new proof for Vector Space of a theorem proved by Frechet in his 1920 paper on abstract polynomials. 2..Z. Theorem (13.31) A necessary and sufficient condition that a function p(A) on R t ·o E( R) be a polynomi a l of degree n is that it satisfy the three conditions. (1) p()\ ) is continuous. '\ '\· 'I-( 2 ) ~or all 6 /\ • • • , .._ n+l 1 , and. all \, in R we have (n+l ) ( \\ A p N =0 ( 3) For some A }. '-l l ' n L\ P( A ) =Io. • •• ' 1i n). and A in R we have Proof: · A. Necessity: Ir p( \ ) is a polynomial then ( 1) It is continuous by T:~eorem (13.201) (2) Using theorems (12.62) and (12.64), end writing we have n+l (\' J:.PN= r=O n+l But, as is well known ( 6 Ar ) = 0 if r<'.. n+l; hence n+l A p( ) = o B. Sufficiency: We prove by induction on n that the conditions (1), (2), (3) above imply that p()J is a polynomial. sufficiency is clear. For n = 0 the As sumine the res 1i lt for (n - 1) we have that if p(AJ satisfies the conditions (1), (2), (3), then p(~+ f-L) - p(/\) 1s, for each value ofr-4, a polynomial of degree not greater than (n - 1) in I' , and is for at least one value oft"'- of degree exactly (n - 1) in'>-.. To see this we s ir.rply write ~ = .11 A , together with any other set of increments 1., 2 i.,, • • •,.1 n+{ . Then p( " +1) - p(t) = 6 1p()J ..\+1.t.n···1~o+r) - p(f.~ = 6\cA) = o Moreover , for some set of increm e nts y,½'.\ •••, A n A we have (1) Hence where 8r()A a.re functions on R to E whose e .-act nature has not shown up except that for sor.:ie value of ~ an , ( ')j r) -. i O. In ex,.9res s ion (1), putting"= O, we obtain (2) fror.1 which p() +~. ) - p{i,,) - p(·~ + p(O) = I ! n-1 Lr=l "rar(w / . (3) Observing that the left hand side of (3) is symme tric in~},fwe have (4) By (l~.204) the coefficient 8r(lJ-) may be written as a linear co~bination with constant numerical coefficient of terms obtained by replacing Ain , 0. the right hand side of (4) successively by the values 0,1,2,···,n. Thus it follows that ar(~), being a linear cor~bination of po lynomials in r-1 , is also a polynomial in p- whose degree is at most n-1. Thus n-1 = 7 - r,s=l ---.r \..A.sa r-- 1 (5) rs where ars are fixed elements of E(R). Sup~ose now first that \ chosen so that O <'.'.. f</'i. from j=l to j = 1 i s a p ositive value of Let j be an integ er. Put).= r A. Let~ be in (5) and sum "'l~/1, where \L,b l d e·J n.otes the greatest inieger in .A.l. I ~ tA- - We obtain N0 w since li.'11 t"~ lim r~ \--1 q~- i ~1i z..1 y-> +0 j=l it follows fro m the fact that p("),) is continuous and fro r.: T 13 (12.3), that as r tends to zero fro m the positive side, the ri ght hand side of ( 6) and the fi n :.t t wo term s of the l e ft hand side bave well defined limits; and that, consequently, the las t ter~ o n the left side also tends to a limit wh ich we shall denote by K. p(),_) - p(O) - K = The equation (6) then becomes n-1 A r+l . L l ~l r=l r + 1 K ■ when r ~l 1~o/) lim p.➔ +O t,A- ..J - p( 0)) i l . p{ ~ ) - p(O) ~ lim ~ A _l r ➔ +o - ~ ~ ~ lim Pf-) - n ( 0) r,-> +O ~ = \. Hence, (7) gives forA 1 > 0 (8) If Al is ta£en to be negative, i-1 may also be restricted to be negative. The argument carri e s t hrough in precisely the same ~on and we obtain 'L ) = p( 0) + l · lim P( "'i ,1 f➔ -0 But by writing A = from whi:h lim -rin p-) -0 · n-1 . P( t") - 'P ( O) + Lr:;/" r=l A r+l ::1 ( 9) -r . +-1 1L -r1 (5) and making ~ ➔ +0 we have nq.,.) - p(O) = (10) ~' Thus (8) and (9) coincide for ),1 ! 0 and either (8) or (9) is at once seen to be correct for A 1 = O. The work therefore shows that p(~) is a polynomial of degree at most n. If it were not exactly n, then the n coefficient of An would be zero and hence~ p(~) 0 trhich by assumption = is not true. The proo: is therefore Complete.• 13.4 P0 lynomials .Q.n. ~ to E 1 (A) Definition (13.41): A f1Jllction p(x) ~ 0 on E(A) to E'(A) is called a polyno□ ial if (1) p(x) is continuous at every x. i (2) There exists an integer n such that for each x,y , ( -~ , PO, ;x,y) = p(x + Ay) is a polynomial on A[E2 J to E' of degree ~ n. ✓ • f\ The least integer n sa.atisfying condition (2) is called the degre e of p(/J . If p( ~) E O, we call it the null polynomial on E(A) to E1 (A) and it is regarded as a membei· of the clas ~ of polynomials of degree m, where mis an integer. Theorem (13.420) Proof: fore constant in).. A polynomial p(x) of degree zero is a constant. p(;\x) is a po·lynomial of degre e zero in ~ and there- Hence putting )_ successively eq11al to O and 1 we have p(x) = p(O) • It is worthwhile to mention here that the exact reason a similar theorem cannot be proven by a similar method for polynomials on C to E(C) is that t he limit in (10) is in that case not necessarily unique. If to the conditions (1), (2), (3), we added the conditiono~p(A) of having a differential at~ = 0 the theorem just proved could be proved for polynomials C to E(C). To show that the theorem is not nece s sarily true for polynomials C to E(C), observe that p(A) ~ ~ , the complex conjugate of A , satisfies the conditions (1), (2), (3) for n = 1 and is not a polynomial in the sense of (13 .2) i"' ..,,_ ,_ •• '. i •'· L Analogous to Theorem (13.203) we have: Theorem (13.421) Let p 1 (x),p2 (x),•·•,pJx) be m polyno□ ials of degree ~ n on E(A) to E' (A). m T'nen any linear combination p(x) =~Pr (x) with coefficients from A is a polynomial on E(A) to E'(A) whose degree is at most n. Proof: tinuous . p(x), being the sum of continuous functions, is con- Condition (1) of the definition (13.41) is therefore satisfied. Then P1 (A;x,y) is, by (13.41) and the hypotheses, for each x,y a polynomial in" of degree at n:ost n. Hence by Theorem (13.203) m p(x +X y) = 2ex rP 1 ( ~;x,y ) r=O is a poly aomial of degree at most n in A. The second condition is there- :ore satisfied. Tri eorem (13.422) to E'(A). Let p (x ) be a polynomial of degree n on E(A ;, Let c be a f i xed nurrb er from A, dHferent from zero, and x 0 be a fixed element of E(A). Then p(x) a p(x0 + ex) is a polynomial of degree n in x. Proof: ~o + ex is clearly a continuous function on E(A) to E(A). p(x) = p(x 0 + ex) ix: there f ore a conb.inuous function on E to E1 of a function continuous on E to E which, by (12.54), must be continuous. Again , p(x + )..y) = p(x 0 +ex+ /\ cy) = PO,. ;x0 + cx,cy), which is a polynomial in~. Let p(X) be a polynomial of degree non E to E'. Theorem (13.423) n Let P()_;x,y) =p(x+1-y) =L )...r k (x,y). Then for each r the function r=O r i;.(x,y) is u.:1i que and is a polynomial in x for each y and in y for each x•. Proof: (13.204). Let Ars (r,s = O,l,··•,n) be the numbers defined in Gsing that theorem we have Is-(x,y) = S=O Now by (13 .422) fixe :i x. Ar 8 p(x + sy) p(x + sy) is a polynomial in x for fixed y and in y for By Theo rem (13.421) it follows that kx(x,y ) is a polynomial in x for fixed. y and a :polynomial in y for fixed x. Definition: A function h(x) on E(A) to E'(A) is called homo- geneous of degree n, if there exists an integer n such that for all x of E and all A of A we have h(Ax) = )..~(x) Theorer., (13.424) I f h(x) is a polynomial on E(A) to E 1 4A) and is a homo 6 eneous function of degr ee n, then it is a polynomial of degree n. Proof: If h(x) ■ O, the theorem is true by definition. If h(x) :I 0, it must have some degree N. ·,·.- e must prove that 11 = n. By the definition of a polynomial of degree N we have N h(x + Ay) = ~}._r ~(x, y) 0 where Is-(x,y) is, ac cording to Theor em (13 .423), a uniquely deterITined polynomial in each of x and y. If in ( l) we place x-=O, then ( l) must reduce to the known identity *A more ex.act statement as to the nature of kr(x,y) is given E. little later; see Theorem (13 .428) (1) Hence it follows that kn(O,y) = h(y) and therefore that N ~ n. N > n. 7: e A8 surne have ( ~ J O) Eouatin 6 t he coefficients of)! in (1) and (3), we obtain (4) Now if N > n, the eA'3) 0nent of Unless, therefore , ~( x,y) r on the ri ght hand side of (4) is negative. = O, e quation (4) is a contradiction of the fact that ~~(x,y) is a polynomial. 'i'; e are, by t h i s last theorerr. , justified in sp eaking either of a "polynomial homogeneous of degree n 11 or of a "homogen eou s polynomial of degr e e n 11 • Theorem ( 13 .425) If h(x) is a ho~rogen eous polynomial of degree n, and if kr(x,y) is def inel ~s in the ~rededing t wo theorems, then kr(x,y) is homogeneous of de gre e r in y a.nd of degree n-r in x. ~: Let n kz.(x,y) A ~ , ')._ .r (1) then h(x + ,. y) = h( x + µ µ y) = L ( - . \ ~(X.JAY) I \ r::O \ J I (2) and h(x + f. y) (3) I"" Equating coefficients of )__r in (1) ani (2) we obtain 3o. and doing a si milar thing for the coefficients of )..r in (1) and (3), Theorem (13.426) Let h(x) = h 1 (x) + ··· + ~(x) be the sum of m homogeneous polynomi ~ls of degree n. Then h(x) is a polynomial homo- geneous of degr ee n. Proof: h(x) is clearl y homogeneous; for, It is also a polynomial by (13 .421). Theorem (1 3 ,427) A polynomial p(x) of degree n is uniquely re- presentable as a sum of howof eneous polynomials of degrees~ n Proof: By taking x = 0 in (13.423) we obtain (1) and (2) therefore (3) Placing -,_ = 1 in (1) we have p (y) = 2 hr(y) (4) r=O Suppose now thnt we bad another representation of p(y) as a sum of homogen eous polynomials. Let it be 3 1. where n.r(y) is some polyn omial ho~ogeneous of degree r. TGen for all we have By theo rem (13 .206) we have hr(y) = br(y). The apparent abundancy of terms in (5) i s taken care of by observing that if N) n, p(y) cannot satisfy the d efi nition for a polynomial of degree n. Theorem (13 .428) let p(x +).y) = Let p(x) be a polynomial of degree. n. As befor e n r Z°X ~(x,y). Then kr(x,y) is a polynomial of degree n-r rdJ in x and is homogeneous of degree r in y. Proof: By (13 .427) we have n p(x) = L ~(x) (1) r=O an d b y theorem (1:: .425) v:e have · r hr(x +).y) = L )_ s krs(x,y) (2) s=O where k:t- 8 (x,y) is of degree s in y and. r-s in x. n Hence p(x + }.y) = L r=O (3) - n ,-. k rs (x,y) T~e coefficient of )..8 is by (13.426) homogeneous of degrees in y and by ·(13.421) of degree= (n-r) in x. This compl et es the proof. .3 2. 13.5 Characterization of Polynomials QB. E(R) to E 1 (R) 13.51 The Frechet Definition The definition given by FTechet of an abstract polynomial on an E(R) to an E•(~) is the following : ~ function p(x) on E(R) to E 1 (R) is ce.J.led a polynomial of (1) It is continuous at each x . (2) For arbitrary increments"61x,D2 x,••·, t.n+lx and arbitrary x we have n+l t:. r,(x) - o ( 3) For some set of ir..crerr - nt s have a1 x , •••, I:.. nx and some x we n 6 p( x) /. o• We propose now to prove two things : first , that any poly- nomial of de gree non E(A) to E'(A) i~ the sense of (1 3 .41) sati sfies the con._1_i t ions ( 1), ( 2) , and ( 3) ab ,.:,ve whether or not A = R; sec : nd) that the above definition of polynorr:ials on E(R) to E'(R) implies the defi nition (12 . 41) for the special case A= R. For c Jnvenience we shall refer to a function satisfying (1), (2), and (3) as a Frechet polynomial. The word 11 polynomiaP, alone, will refer to the definition of (13.41). ~eorem (13.521) to E' ( A). Let p(x) be a polynomial of degree non E(A) ThenL1p(x) is for all choices of6x a polynomial in x of d. egree ~ n-1, and for proper selection of Ax a non-null polynomial of deg ree exactly n-1 . • The notation is that explained in (12 . 6). By application of Theorem (13 .428) wc ose equations Proof: we shall here assume, we have p(x + Ax) = p(6x +Ax) / A= 1 where Ir:r<Ax,x) is of degree r in x and=. n-r in~x. Since ku_(Ax,x) is of degree O in Ax we have from ( 13 .420) that (2) Therefore is the sum of polynomials of degree~ n-1 i n x, which by (13 .42~ is a polynomiel of degree ~ n-1. To show the second part of the theorem we write p(x) as a sum of horr.o6 ene ous :polynomials . so that t. p(/'-X) = i Theorem (13.427) hr(Ax +C. x) - hy-0.x) If we write~ x = x, t his becomes (3) so that the coefficient of ,e-l in p()-.x) when 6x =xis which, s i. nce 1¾i(x) ~ 0, is a non-null polynomial of degr ee n. Let x 1 be a value for which it does not van ish. Then with x = Xi., the coeffici ent of '}..n-1;:'W\the 8X!)S..Ylsion of ~-p().x) is not null since it does not vanish in particular for x = x • 1 chosen.6 x, Thus for suitably A p(x) is a polynomial of degree exactly n-1. Theorem (13.522) If p(x) is a polynomial of degre e non E(A) to E'(A), then for all ~x,~x,·•·, ~+1x, we have Proof: If Theorem (13.521) is applied succ essively to L i:p(x),~2~(x),•··, then it follo ws that~-·· L 1p(x) is of degree zero at rrost, and is therefore cora tant in x. n+l ~ p ( X) ~ Theorem (13 .523) Hence 4i •••J11 X + 'it+l X) - ~n • • • '1 -p ( X) = 0 . !) ( If p (x) is a polynomial of degre e non $( A) to E' (A) then there exists a set of increments /j 1 x, ~x, · • · ,Cnx s uch that Proof: We use the second ?>art of Theorem (13.521), select &:; 1 x so that ~l p(x) is a non-null polynon-,ial of degree exactly n-1 in x. Select6 2 x so that 6 2'1.p(x) is a non-null polynomial of degre e exactly n-2, and so on. With this sele ction we must havelp(x) ~ 0 Combining now the results of the last two theorems we have: Theorem (13 .53) If p(x) is a polynomial of degree non E(A) to E' (A), then it is a Frechet polynomial of degre e n. Proof: (1) A polynomial is cont inuous and therefore satisf ies the Frechet condition (1). ( ) ( 2 ) By Theore:n ( 13.522 1, n+l ~ p x = 0 for all sets of increments. (3) By Theorem (13.523) there exists a set of increments l 1x, • • ·, D.nx for which /).np( x) ,; O. We shall now prove a converse for spaces E(R). Before pro- Geeding, let us prove the following Lemma. Lemma. (13 ,540) . If f(A) is a continuous function on R to E(R), ~1 satisfying the condition '1 f(A.) A' = 0 for all choices of 7.A-, A,?·,•·•, n"- then f(; _) is a polynomial on R to E(R) of degre e at most n. Proof: Let m be the maY.imurn integer for which there exists a ~,4:?,,···,~A such that ~mf (; ) ~ 0. cho ice of A1 n+l * since f),; f( ) ) Clearly m ~ n; for, = O, -all differences of order greater than n mu.st van i sh i de~tically. Then by the defi nition of m, for all choices of the.6 1 s m+l we must have f:... f(A-) = O. Therefore by Theorem (13 .31), f(A) is a poly- no~ial of degree m ~ n. Theorem Cl;~.54) If p(x) is a Frechet poly!10T!lial of degre e n on E(R) to L 1 (R) then it is a polynomial of degree non E( R) to E 1 (R) in the sense of ( 13.41). Proof: g()_) _ q(x +A y). Let q(x) be any function on E(R) to E'(R). Let A Abe any increment of A and define Let 4 g().. ) - q <x + ). y + L1 , y) - q <x +; y) Then - ...: q( X + I y) Now let (1) = p ( X + Ay) . f ( /' ) Let an arbitrary set of increments >1x =.:_?· •y (i = 1,2, ·· •,n+l). and let.... Li),~\··•,-~J,1 be chosen Then applying the formula (1) successively to ~f(l) : L1 p(x +Ay) 1 D we obtain n+l f ("') .) 2 u1 f( A) = LJ2 l 1p(x +\ y) = n+l /. -p(x + /" y ) • N0 w f( A) is evidently a function on •-~ R to E'(R) and is continuous since p(x) is continuous. Hence, a~~lying Lemma (13 .540), we hB.ve the existence of m = n such that f( ; .) is a polynomial of degree m on R to E'(R). p(x ) is therefore a polynomi cl of degr e e m in the sense of (13.41). But m cannot be less than n; for, by theorem (13 .522), we should have for all !J 1 x,~x,•••, ~ i 1p(x) ~ 0, and hence have thatL1np(x) = O. x, 1 Th is would contradict the t hird Frechet condi tion. This last work may be surrrned up in the theorem: Theorem (13.55) For spaces E(R), E'(R) the definitions (13.41) and (13.51) of a pol~momi al of degree n are equivalent. 13. 6 Homogeneous Pol:x-nordals and their Po:L:t rs. Definition {13 .61) non E(A) to E'(A). ./.ul ti linear f onns. Let h(x) be a homogeneous polynomial of degree Then the function h(x1 ,•••,Xn_) on En(A) to E'(A) n defined as .Q_ h( 0), where /J. ix x 1 wi 11 be called the complete polar or n1 simply the polar of h(x) with respect to x 1 ,···,Xa· = Theorem (13.62) If h(~,··•,xn) is the polar of h(x), then in each of its arguments it is a linear homogeneous polynomial . It is, further- ~ore, syrrmetric in the arguments. The symrretry o,f j1h(o) in the increrr.ents was noted in hQ_Qf: ( 12 .63). nth. It is therefore sufficient to consider one of t hem, say the By n-1 successive applications of !heorem (13.521), it is easily proved as in (13.521) that ~1ri (x) is a polynomial in x of degre e or.e, which, for suitable ci.1 oic:es of~~···, f,x) ~-!' is non-null. Therefore n-1 = n-1 l h(x) - L.1 h(O). is a polynomial of degree one on Ett1- 1 J to E'. Hence we may write f()..x)= k (x) + k 1 (x) 0 by (13.427) (1) (2) Placing x = 0 i n (1) we obtain f(O) = 0, from which ko(x) = 0, ~(x) = f(x). Thus (2) yields But f(x) is by its definitionL11\i(o) where Ll nx = x. Definition: Any fonn which, like h(xi,···,~), is linear homogeneous and continuous in each of its arguments is called a multi- linear form. The nolar b ~ars to the form a relation analogcus to that which holds for ordinary al geb raic forms. Theorem (13.63) Let h(x) be a homogeneous polynomial of degree n, and let h(x 1 , · • · ,x ) be its polar . 0 This is given by Then h(x) = h(x,x, • • • ,x). t 1h(O) = h(x) - h(O) Then 2 ~ h(O) = h(2x) - 2h(x) + h(O) This is a well lmown interpolation fonnula wr_osE- ~t~ -be.i'tJv.\ is Using now tee homogeneity of h(x) ~ n-r n 6n h(O) = '(-) (r)(n - r)nh(x) = n ! h(x) h(x) = 2,h(O) = h(x,x,···,x) or n! Conversely we have: Theorem (13 .64) Fora given nomogeneous polynomial h(x) of degree n, there exists no symmetric multilinear form other tha n the polar of h(x) which enjoys the property described in (13.63). mul t ilinear forms and let h( x) be a ho1:-1og eneous polynomial of de g ree n such that h(x) = h(x,x,•••,x) = h(x,x, ··•,x) (1) We shall em~loy induction on n to prove that In ( 1) put x = x + A y, and make use of ( 13 .425): and the distributivity of h(x,••·,x) and h(x,·••,x) with respect to their arguments. This gives. where ~(x,y) is a polynomial ho~ogeneous of degree r in y and n-r in x, and where ~(x,y) and ~(x,y) ar e the functions obtained by writing = X respec ti vely. n-r = X Equatin6 the coefficients of \', we obtain lq(x,y) = n h(x,x,•··,x,y) = n h(x,x,•••,x,h) (3) A.ssumint~ the theorem true for (n-1 ) we have for any value of y that the two forms hand h ar 0 syrr.rr'. etric multilinear forms in (n-1) arguments x and are equal to a homoge neous polynomial of degree n-1 in x. Therefore To complet ~ the induc t i on we observe that the theorem is trivial for n:::l.• A general r esult on multilinear for n s for complete spaces wil l be derived in a number of theor·ems. Theorem {13.65) Hypotheses : H1. Let x be a typical el i:;ment of a vector space E1 • ¾· Let y be a typical element of a complete vector space E2. ¾· 1 et B(x,y) be continuous on Ei_[E2Jto E' at X = 0. !4. Let B(x,y) be continuous linear homogeneous on [ E1] E2 to E'. • A proof could be given for the above theorem without assuming that h(x,x, .•. ,x) and h(x,x, ... ,x) were each equal to a honogeneous polynomial in x, but it was t hought of too late to be included. 40 , 85· Let B(0,y) = 0. B(x,y) i s continuous on E1 E2 to E' at the point (x,y ) = (0, 0 ) Conclusion : Proof: ·,i e wish to pro\'"e tha.t for E > 0 there exists 8f: > 0 s1.:.ch that if (x,y) is in (0,0) 5~ , the 6~neighborhood of (O,O) >t h en I'I E(x,y) - B(O,o) !\I) == '. ' B(x,y) ii < €. ' 1 1 1 We shall denote the point (0,0) by (0). If the the c., re:r: were not true then we sh ould have k > 0 such_ that in every (0) g- there exists at least one point (x, y ) for wni ch I\ B(x,y) H> k . \',e prove that this h~,pothesis leads to a contradiction. Select an infi ::1 te sequence o f points ~fow (Xu,Yn) a s foll ows : by F and H ,rn may sele ct 8' <. 1/2 suc :1 tnat for ( x,y ) in (o) we have 4 1 0 81 Con~i nue t nis process. cho os e &p-+-l < In gen2r a l, hav i ng sel 2cted(x:...., y) in (O)~ ~¼l:~. s1.:.cn that for (x,y) in (0)~1 }J p Op 4 I. (i =v>≡>∙∙∙>p) (!) II B(x’yi)||‘;2t+V such that then select (x^.y^ P Now let Yp (i =½≡j∙∙∙,p) Since ∣[ ⅞rι ('∣ 4 < 1 /2* we have Since E^, is complete this is a sufficient condition for the existence of a limit Y for the sequence , Yp>∙ Now, since B(x,y) is by Hλ continuou⅞ and linear in y, we have (2) or (3) But by the selection of (x-,y^) in (1) we have II ≡(n>yp l∣< 2j÷k__z (4) αa Hence the limit on the right hand side of (e) is less than z J≡O which is less than k∕2. k 2 ί -‡ 2- ? Remembering that ∣∣^χi>yι)∣l > k we have from (S) ∣( B(xi,Y)∣∣> k - k/2 > k/2 4 2. Since this holds for all i and since lim i ➔ CD of¾· x i = 0, we have a contradiction This proves the theorem. Theorem (13 .66 ) Let E1 ,E2 ,···,En be~ complete vector spaces, Let x. be a represent ative point of Ei . distinct or not . Let 1 -~(x 1 ,~, • · •Xn ) be a. multiline ~Jr for m on E1E 2 • · •En to E 1 • Then },: is con- tiri·..10us at ( x1 , x2, · · · ,Xn,) = (0,0, • • • ,0) -:;: e p1·ove the theorem by induction on n. Proof : theorBm is true by definition . For n=l the I f it is true for n-1, then ..1( x 1 , • • ·, ¾) EiJL.a .. ·~-i LEu] to "~ at (x1 ,¾, •· •,~_ 1 ) = (o,o, ·· · , O) 1 is continuous on and is linear homogeneous and con tinuous on [E1 •E2••• En-L Ezi to E' . Therefore by Theorem (1 3.65) Mis c ontinuous at (x ,¾,·••,x0 ) = (0,0,·••,0), 1 on Ei •E2 ···Ez.i to E' which was to be proved. The spaces ~(a,b) are as i s well known, instance s of co~plete vec tor s'!)8.ces. The result of (1 3 .66 ) and a consequence of it will be a~surr.ed in pa rt 2. 13.7 ·:~odular Prooerties of Poly!10~ials and r ultilinear Forms . Definition : By the n:odulus of a hor:wg eneous poly~:omi al of degree n we shall understand the U?per bound of the expression~ II x1F values of x . for all We denote t hi s bound by mh . To prove that mh exi s t s we have : Theorem (13 .71) then the expression Proof: that there exists If h(x) is a homogeneous polynomi a l of degree n, I\ h( x) I( is bounded for all x. \I x\\n By defi nition ~( x) is continupu.s at x = 0 . It follows o'i such that for 11 xi{<~ ll h(x) l\<. 1. Now if x is 4 . A X we have ar b itrary, and 1f x = ~1-/ 21tx I \( x ri = \\ .8_1-L\\ = tl L._ 0 2f x\\ 2 1 \\ h(x) H< 1 '.I1herefore But Hence The upper -bound mh theref9r e exists. Definition: By t h e modulus of a mul tiiiinear form M(x 1 ,x. ), · · • ,~) vie shall understand the up ~er bound , in case it exists, c.., of the expression // Mj x1.,J1;.~ • • • llJ over all values of the x' s. I\ / xlfl •IT x2(i • · •I ~ It is -~ enoted by mnM . Theorem (13.72) I f E1 ,E2 , ···,En ar e com .,let e ve ctor s paces, is bounde,j over all x' s. Proof : By theorer:: ( 13 .66) L is co ntinuous at the :point (al'x 2 , • · ·, ~) = (0,0, · · · ,0). if II xd < 81 ( i = 1, 2 , •. , n) . •- 0 X anQ~ if x 1 = -1~- 2 1( xi II \Thenc e There exists, therefore, t h en 11 MI\ <. l. we have as before 0'1, suc11 tha t If x1 , ¾, • • •,~ ar e arbitrary j It follows that IDn.i.: exists. If h x) is a nornog eneous :polynor.-1ial of degree n, Theorem (l e . 721) and if I( x \11... r then rJ h( x) Proof : l'it ,n ( x;' Theorem (l ~ .722) \\ xi\k ri \I <. rr.hrn. ·, ,1 1 1 / mh rn m h !.x ', 1n -.. I f: : is a multilinea r f orm i n ~' • · • .,xn' if (i = 1,2, • •· ,n), and if mnM exists, then l\M lk rr1uM•r 1 -r2 • • •rn· Proof: Similar to (1 3 .721) Theormri (l e . 728) If h( x) is a h or:;ogeneous p olynomial of degree n, then Proof : (1) since the second maximum is taken over a more restricted set of x•s. Now let x 0 be a va lue for which \ h(x 0 ) Ii ,/ · max ;r1 h(flY',, II x1;;n it xtl =r Th en put x 0 (2) r so that 1) i 0 1( = r )' Thus I/ h(x) 11 max r1l 11xll =r Thi s contraci.iction proves t hat the ine quality in (1) is i na drnis sable. r r U V e cl ~ ; V"1 ,· ( C.. r { '( j (C. '-> I I c, W S t VY\ -.,..,. e .:J 1' C.. 1- (. / y f r VY\ ('o._ ) C.... V\. d l O/ lb \ \ 5 J Theorem ( 12 . 724) ;: L) If ~: (x 1 , • • • ; Xn) is a mul tilinear form continuous at (O,O,•••,O) then IDn.M exists. Proof: An exact reproduction of (13 .72); tne only a s sumptions use d there_ in the proof we re the hypotheses of the present theorem. Theorem (l Z.725) I f M(x1 , •• • ,~) is a multilinear for m s uch that IInM exists, then M Proof: Precisely s ird lar to (13 .723) We shall now prov l?. a special t heorem on the modulus of the polar of a ho:. o6 ene ous pol ynomial. Theorem ( 13 . 73) of d.e 6 ree n. Let rnh be the r:1 od ul us of a horr: og eneous polynomial Then the modulus IDnh of its polar exists and we have the relation Proof: Let ~ x • •. '1 x 1 , ' n Since n ~ n 6 h(x) is of deg ree O Li x, we have 6. h(x) = 6 h(O). be an arbitrary set of incren ents and take 1 Consi der t he manner in wh i ch the succe ~. s i ve differences of n( x ) are formed (1) At each st age , eac11 of the terms on t he ri ght nand side r ive s rise to two (2) new ones. Ah(x) is t herefo r e the sum of 2n terms of t he type where e 1 = ±1. 0 () But (2) ll Therefore th(0) /1 :! ''I)B. h (,x) ,/'1 ~ ~, mh ,1.' ~ ) n • ( max /'1~). x I\)n 1 n! n. n. 2 (3) 1.,, The existence of mnh follows at once fro n: ( 3 ) and Theorem (l c .72A), for ( 3 ) s i10w2. t hat n (x1 ,x2 , • • • ,xn) is co ntinu ous a t (0,0, • • · ,O). If in ( 3 ) v;e re qui re thatlk6 1 x l\ = 1 (i = 1, 2, ... ,n), then ( 3 ) !I 2' h(O) {! ~ mh 'fln becomes n! (4) n! By theorem (13 .725) t h erefore ffinh = max I\ Aj_ x l/=l Ii i:2 h( o) I\ ~ mh an n! (5) n! (6) Again we have (lomb ining ~5) and (6) we obtain (7) wnich was t o b e n rove l . As a corol lary to the last theorem we have. Theorem ( 13 . 72;1) A nece ssary and sufficient condition that a h omogeneous polynomi al be null i s that its pola r be null. Proof: Ey esuat ion (?) of Theorem (1 3 .73 ) mh and mnh mus t vanish t ogether. \ie ha ve now a special result for spaces E(C) whose possibi lity s e ~ms to depend upon the f act that in the complex plane the functions z,z 2 ,··•,z n , are orthogonal on t he unit circle to the set wi t h respe ct to th e function 1/z Z -1 z -2 • • • z-n , , ) Theorem ( 13, 74) n Let P(1') ~ L { ar be a polynomial on C · to E( C) 0 Th en for alls (s = 1,2, •.. ,n) Proof: Let h , k , s b e in teg ers "\ f\ h s ~ n. For c onvenience def i ne ~ 2TTifn - e Th en Now define ( r = s) 11m So that h ➔ oo ( r / a) Def ine firom which :But Hence ~i n&lly (( as \I, T I' < ' ( ' I\ = 11ilim Uhs Ii:\ = Um Ii' ½is l! = max :j p /.I / h ➔ co h ➔ oo I l=l Theorem (1 3.?S) E'(C). Let p(x) be a polynomial of det:7ee non E(C) to Let it be represented as a · sum of ho~ogeneous polynomials in t h e 48, form p(x) = t hr(x), where ~(x) is of degree r. mhr ~ Then for all r max {\ p(x) II 11Xf1=1 Let \} x II= 1 -Write Then by Th eorer:1 ( 13 . 74) \l hr(x) 11 max 1\ p(M)!(~ max 1\p(x) \[ ,; ll/\ \i=l (1) I\X//=1 Since t ni s holds for a ll x for which II x ll = 1 ~ = maxi(~ max p(x) ~ x''=l px(f=l (2) 1 1 1 which was to b e proved. These last two theorems are of some use in the theory of analytic fun ction s to be intr oduced in the next division. 13.8 Differentials of Polynomials Di ff erentials were def i ~ed in (12.6). I n t his section we pro- pose to show that :-, polynomial p oseeses dif f erentials of a ll or ders and that t he differentials are syrranetric in the i ncrements. Further~ore, it wi ll be shown that the differenti 2ls of a homog eneous polynomial may te convenien tly ex-~ res sed in t erms of its polar. At this point it is perhaps not rur. iss to prove the re s-dt we refrai ned. fro :,: proving in section (12. o ). Theorem ( 13.80) Let f 1 (x), f (x), ... , fm(x) be functions on E0 2 to E' dif:·erentiable at a point x 0 of E0 • m m Then f(x) differentiabl e at Xo, and d.f(x 0 ) = ~ dfr(x0 ). r=O = ~ fr(x) is Proof: df(x 0 ) is the sum of lines r homogeneous polynomials in ~ x and 1s ther efore also linear ho~ogeneous. fr(x 0 ) - dfr(x 0 ) !I 6 X I\ Let so ~hat b y t he definition of dfr( x 0 ), E-r(Lh)-., 0 with 6. x . Therefore as /ltix II ~ O € (ti x) ➔ O and Theorem (13.81) If p(x) is a polyno~ial of ~egr ee n on E to E', t h en at every x of E the diLerential dn(x) exists. Ern:9loying Theorem (13.428) v;e write p (x + ,A y) = L..B... ~r ~(x,y) (1) r=O where kr(x,y) is hoGogene ous of degree r in y and of degree~ n-r in x. Viriting J.. succes d vely equal to O and 1 v: e obtain. (2) n p(x + y) = L. (3) kr(x,y) r=O We proyose to s ~ow that if we take Ll x ~ y, then the differenti al of p (x) is p recisely k 1 ( x ,!bc ) wh ich is c. linear function of fix. 1 Combining ( 2 ) and (3) and rearr anging n p(x +flx) - p( x) - k1( x ,AX) = Z,., ~(x,A.x) r=i:::; (4) Regarding x as fixed we have II p(x + Ll x) - p(x) - ~ ( r., h. x) /( ~ fu m ~(x) •{~ x ;f n This is in exactl:r the form of ( 12. 6 ) As\! D. x \l, ➔ 0 the ri ght hand member te:id s to zero :::nd t herefore so als o does~(-Ax). Theorem (1 3 .82) d.h(x) If h~x) is a po l ynomial homoge neous of degree n t hen = hx(x;Ax) is given in terms of the nolar of h by nh(x,x, •·•,x,6x). Proof: Employ the e xpansion us e ' in Theorem (13 .64) e ~uation (2) VTi th L x writ ten i n p lace of y. l' sins the Ga.4teaus met nod of calculating dh(x) we have = n h1(x,6..x) = n h( x .,x,•••,x,Ax). Theorer:, (13.83) h(x) of degree n (r ~ n The r th dif f erentia l of a homogeneous polynomi al is given b y n! (_' n - r ) •I Pr oof: h ( LJA l.' X • • • .1 X X X • • • X) > r > J ~, ' :5'or r = 1 t h is is Theore.:, (1 ~ . 82). true for r - 1, then If the theorei:: is - d n! " d ~'( h x) = r-1 d r- 2··· d 1h ( x ) = -(n-----'-r-'-+-1) ! h(6lx,••·, 'i--lx,x,•••,x) (1) i s a horr.og eneous polyno:T: i nl of deg ree ( n - r + 1) and the exc:,r e ssion on the ri ght of (1), be ing a r..ultilinear form, r.iust be p r ecisely its polar. Therefore we may apgl y (13.82) agaL to (1) and obtai n the result for r. This completes an induction on r. An i mr.1ediate consequence of the las t theorem is. Theorem (l o .84) The r th differential of a polynomial is sy:-~~etric i n the inc rement s. Proof: From the symmetry of the polar of a -hor..ogeneous pol~ no:-:~ ial i n its increments, and from 'Theorem (1 :3 .83) follows the symmetry in ·d i fferentials. :But since an~, polynomi &l is tne sum of homog eneous poly- non ials a nd since the surr. of the di: : erentia ls i s the di:fferentL-1 of the sum, th e symr:: etry holi_;_s for 13 . 9 cn~,r polynorn ial. Polvnornia ls of Polynomials. Polyno·~.ials Q.D.. Comoosit e Spaces. I -:1 this section we sha l l discuss rather briefly hiO sit1.1c:.tions in which we shc:. 11 b e i n terested in the next divisi on. :siLst , let q(x J ce a po l yr.i.omial on E(A) to E' (A) and p(y) be a 1:)0lynomial on E' (A) to E11 (A). P(:x) = p(q(x)). Le·t us consider the nat ure of the function ',','e shall sh ow t:-:a t F(x) i~ a polynomial on E(A) to E11 (.A) and that its deg ree is at mo st m•n, wher e mis the deg ree of p and. n is that of q. Second, we snr.11 exar:·: ine the state of af f air ~ when polynomials on a composite s nace E1 Ez to E' are reg arded a s on E{. E2~ to E'. To pro ceed with the first part, let us prove: ·,t.:' Theorem (13.91) Let p(x) be of degree non E(A) to E1 (A). z( ~ ) be a polynomial of degree m on A to E(A). Then P(~) Let = p(z) is a p olynomial of deg ree at most m•n on A to E ' (A). Proof: and true . :le use induction ovei~ m. If m = O, the th e orerr. is trivial Referring to (l~.423 ) we have z r=O n p(x + y) = ( 1) kr(x,y) where k ( x , y ) i s of !}",gree = (n - r) i n x and homog :meous of degree r in y. r Let As sum e th s theor em for all polynomials z of degree~ m. by { 8ni· Re:9lace yin (1) This yields m-1 I f xis now re~lace~ byL A 8 as then, under the induction hypothesis ~ S=O kz.( x ,¾-i) be cor::ies a p olynomial of de g re e a t rnost(m - l)( n - r). I f this :9o l yno:;; i a l is multi:Jlied by Amr., its de g ree will b e at most mr + (m - l)(n - r) = mn - (n - r) ~ ~n The r e fore P ()- ) = p ( z) fr th e Th eorem (l ~.92) S UD Let q(x) be of d -gre e rn on E(A) to E'(A) and p(u) o:· degree n on E ' (A) to E 11 (A). at E:O ~ t of p olynomials -of deg rees ~ m• n. Then P(x) = p( q ( x) ) is a p olynomial of deg ree m- n on E(A) to E 11 (A). Proof: J?(x) is a conti nu ous fun c tion of a continuous function, and the yefore continuous. It remains t o prove that P(x +f y) is a polynomial S".3 . of degree ~ n•m in .l. . l~ow q(x + ). y) i s for fi xed x,y a ~Jo l ynomi a l of deg ree ~ m on A to E' ( A). Hence ap 9l y i ng Theore□ ( 13 . 91) P (x +) y ) = p(q(x + ). y )) is of degr e e at r:i os t rn •n in ~ . The theorem is t h e ref or e p ro v ed . Le t u s no ¥1 con side r a comp osi :e s·-: ace E1 E2 ( .A ), of t wo spaces E1 (A) a nd ½( A). ·,•: e s11dl d sn ot e e l 2rr:e nt s of E 1 e h me nts of E2 by letters x,y,z, · · ., . b y l e t t ers x , y ,z _, •··, The p oint ( x ,x) of the CO r:l ·J OSit e sup-port we sha ll den ote by a ca -J i ta l letter X. 1 The orem (1 3.93 ) If P(X) is a polyno~i a l of deg ree non E1E2 (A) to E'(A), th en ·9(x) = P(X) is a p olynomial i n x on E1[E21 to E'(A). Proof: By Th8 orern ( I Z. . S-3), ?( X) is con tinuous on Ei [E;z] to E 1 • How b y the definition of a p olynomi a l, P(X +)... Y) is a :pol ynomial i n)_ of d eg ree ~ n on ·4 E1 E;a)'2-]to E '. But if Y:: (y, O) P(X + )_ Y) = P[( x,x)+ >- (y,CDJ = p [(x + >. Y, X + t o)] = p(x + ). Y) The refo r e p ( x + A y ) is a po l yn omi a l of d eg r ee ~ n in}.. argur., en t v: e ~ n on LE11 By a s i milar coul d have p rove d. t h at p( x) ~ P (X) is a p olynor.1 i 2.l of degre e Ez to E' • .., ✓ 14. Analytic Functions in Vector Sn ace 14.1 Introduction I n t his division of the wor ~ we propose to make investi g tions o f· the cons :> quences of a cert dn forr:1a l generalization to v e ctor s-paces of t h e notion of an ana l y tic function of & real or cor:i-· lex variable. .,.:i. nu:-:-:be::.~ of s tudie s have been made i n t i1 is direction*, but rrios tly fro ~ the stand ~oint o f r 8mai :·der theorem s . ..e a U errip t here to 2ttack the pr ob lerr fro ::. wh at is 1 often c a lled the "Weierst r as s viewpoint 11 • ·. . e s hel l de:·ine anal y tic f1.nctions in terr~~ of a "power serie s 11 der lo~ent in hor.-!o: eneo1Js p olynomi a l s and. seek to deri y e the -properti es merely fro r th 1~ consid.eration ~ f inequalities and ident\~s am ong these polynomials. At the end, we shall ap~ly s ome of our results to a diff erenti a l equation theorem. 14 .2 Convergence We she ll in _f:" eneral ~ot re qui re that d 1 spaces with v.':-_ich we de&l obey the :) Os t u l a te c of corn})letene s._. 1 \Then corr. -Jletenes s is ass u re d i n the p:::·esen: section it will be so s pec ified in the theor er.1 . The defi nitio n of the l im it of a sequence of p oints of as ·. ace E was g; iven i n (1 2 .3). ~- other w-y of stating the same thing is that a sequence l Xn_} has a lLLit if there exists in Ea point x such that I( x - Xn \( tends to zero with 1/n. Using the same language , the p ostu late of com~l et enes s requires that the n e ces -· ary and suf f ici ent condition for the e:-'.istence of the limit of a se quence (t Xn} is that ma.x II~Xn+p - xn \1'I teni to zero with 1/n p>O * Se e,for exampl e : L. l! . Graves, Trans . Arn. i.: ath. Soc., V. .-- a ,.:.,-' J 192,- , p 163 • (14.21) C'6Snvergent Seq·uences of Functions The terms Xn of a sequence may depend L 1 various · ay s u ·:·on some gene ral parameter t 'N hose rang e is 1:1 . x 1 (t) In this ca~, e we s ay that the sequence converg es over T if it conver ge s for every fixed value oft in T. (14.23) Uniform Convergence A sequence { Xn ( t) 1depe nding u-;jon a par am et er t whos e rang e is T is said to converge u.nifonJl y int over T if there exists x(t) such t h at max II ¾_( t) - x( t) (I tends to zero with 1/n. t in T (1 4 .24) Convergence of Series n Let { ~ \ be a sequence of elements of E . Let Sn • ; Then if Z1 Xr. we say that Z~ converges to S. Sn converges to a limit S If the co~we rgence of { Sn1r is uniform wit h respect to some paran~eter t ove r a range ) T, we say that LXn co nverges u.n iforr.:ly int over T. A series which does not c _nverge is said to diverge. (14 .25) Absolut e Convergence of Series A series l_ ~ is said to ro nverge ab sol ut ely if the num e rical series ZJ/ Xn}( is convergen t in the ordinary sense. Theorem ( 14.261) Let E0 be a region of E. Let [ fn.( x) J be a se- que nc e of continuous functions on E to E1 converg i ng un iformly over E0 • 0 Then f(x) is continuous over E0 . ~: Sel ect e) 0 and n 0 so that If fn (x) - f(x) \\ ~ ef3 0 Select 8' so that if \( x - x 0 { ( ~ S then Then for all x in (x 0 )~ {( fn (x 0 ) 0 - fn/x) I\ l..e/3 I/ f(x) - f( x0 )j( ~ j/ f(x) - fn 0 (x) ~ + ~ fn (x) - f n 0 ( xJ /1+)[ fllo ( x0 ) - f(x 0 ) \I 0 <. E:. wh ich is the condition for continuity. Theorem (14 .262) i s such t hat If E is co mplet e, and if a sequence ;. Xn ~., from E L ~ i s absolut el y c onvergent, t hen Z' xn is converge nt. n Proo f : Let Sn ■ ~ ~ 1 n+u I\ Sn+p - Sn\\ =~ II ~ II Then but s ince t he ri ght ha nd side is t he partial r emai nder of a c onvergent seri es of positiv~ tenns its max ir:ru.~ value for all p tends to zero 1{ith 1 / n. Then si nce Eis complete t h is is a sufficient condition on Sn+p - Sn for convergence. T:_eorem ( 14.263) Let f (x) n E0 of E to "F: 1 , whe re E' L ~ complet e . over E0 • Then i f LL~ converges, be a s equ enc e of functions on a r egion Let Mn denote t he ma xi mum of fn(x ) fn(x) c onverge s u.nifonnl y i n x over E0 • Let (1) Then Thus Sn( x) converge s to S(x). lor ollary: If the seri e s~ fn( x ) is known to be converg ent · ave~ E0 , then wh ether or not E' is complete the conclusions of Theorem (14 .263) hold. Proof: Completenes s of E' is us ed only to sho..-: convergence If we calcula te t he li mit of ( 1) as i ➔ OO, n remaini ng f ixed Vi e have (2) :rhe ri ght h and sid ' tends to zer o with 1/n and is i ndepe ndent of x. Theorem (14 .264) I f ~ fn(x) ~and ~~(x) r are sequences of functions lbn E0 of E to E' such that L. fn(x) and 2=. g11 (x) c cmverge; then L t fn(x) + ~(x) ~ co nverges and i s e ~ual to Lfn(x) + ~ ~(x). n n P r oof : Let Snf = L f (x) Sng = Z gn(x) 1 1 n ·n Sn(f + g ) = L_ (f ( x) + g (x)) n n 1 T'nen /I Sn(f + g) - / .Zfn ( x) + 'f.gn(x) ) = i\snf - [fn(x) II l\ (1) + Sng - [ f½(x) :; ':rne f ac t t ha · ~ fn and [ ~ ~o n ~erge s hows th.a '._ the right si de of ( 1) t en i s to zero wit h 1 /n It fo llow~- by induction that the sum of any f i.. ite numb er of converg ent seri es is also co nvergent. Theorem ( 14 .265) L ~t Then if~ is a p oint of A, I fn ( x) b e c onvergent on E of E to E' (A). 0 ,Zctfn ( x ) co nverg e s and i s equal t o c;;,1.J~ fn ( x). The p roof is clear. 14.3 Definition of Analytic Functions 2B, ! i£ ~Definition of Regularity : A function Some Definitions f(x) on E, a region of 0 E, to E1 is said to be regular at a point x 0 of E0 if there exists (1) a n ositive number rand (2) a se q_ue nce of homogeneous polynomials h (x) n hn ( x ) of de6ree n-o n E to E', such that for lr x - Xo ll~ r, the series CX) T hn(x - x 0 ) conferges to f(x). 0 The maximum value of r satisfying these co nditions we shall c a l 1. the radius of regularity of f(x) a ssociated with the point Xo and shal l denote it by r(x 0 ). Definition · of Analyticity: A function f(x) on E0 , s regi on of E, to E' is said to be analytic at the point x 0 of E if: 0 (1) it is ( 2) t he moduli mhu of the polynomials 1\i(x) satisfy the regular at x 0 • 1 condition O ~ lim n ~ r ( x0 ), (Illllu) _l n ➔ oo We shall define r'( x 0 ) as t he lim n -- >OO (mh ) n n and refer to it as the radius of analyticity of f ( x) associated with the ~Oint x 0 • vlill also be s·t) oken of as 11 a nalytic r'(x0 ) at x 0 f(x) 11 • We shal l c~l so have oc :: asion to -. .: se the quanti t;y r 11 ( x ) define3. _l as lim o (IDn~) n, where IDn11n denot es the modulus of the polar of hu(x). n ➔ oo Y:e shall sr: ea~: of r 11 (x 0 ) as the ra ;ius of absolute anal yticit :i' of f(x) associated with the point x 0 • Concerning t h is las t we bave incident 2lly: Th,, orem (14...:.ll) If f(x) is on E0 to E', arialytic r' (x ) a t x 0 , 0 then r 11 (x 0 ) satisfies the i neq1:a l ity where e is the base of natural logarithms. By Theorem (,:,, 73) Proof: l ~ ~ mhn so that r 11 ( X 0 = lim ) (mn11n) ~ nn n! -~ _1 = -lini (mhn ) n = r, (xo) n-)oo n ➔ oo , .I. 7 again_ ·r 11 (x· O ) = lim n -• (mn½i) n = lim n co (X) (~fii _J, (rruln) n co Let f(x) ~ ~hu(x - x 0 ) be analytic r'(x0 ) at x . Dominants: 0 0 f > 0, a number r satisfying the condition For a given number for all x and all n, will b e c c.lle ~ a p do-:-inant of f at X • 0 The lower bound o f M's satisfy in s t iiis condition we s hc.11 cc:.11 the r.-·-in i ma .. l IP domi r.ant To s h ow that dominants 12 xist we h e ve _l Since that for f < r' ( x <D 0 ) ~ (x 0 ) = lim (m~) n we have fro r: ordinarJ' analysis n co mhn u converge s . There is therefore a great est r1 f term in this series w:·.ose val u.e let us call M. Thu =r :But ~ < ~.! I = mhn = mhnPn f n From wh ich ~~(x) (\ ~ Ml\~!ln ~ lj X;f pn One such positive numb er r.; existing , t h e lower t cu.}id. c e rt a inly exists. As a r.i.at ter of fact it i s not dif ficult to s e ·- t hat the ~: we have just 14.4 Some pron erties of Analytic Functi ons .QI2_ ! to filA}_. Le t us apply t he defin itions o f (14 .3) to the case where E(.A.) is A i t sel f , and F. ' (A) i s a vec : or snace wi ich in this secti on v.e sh~: 1 1 wr i t e with out the prime . The most l ene ral homogeneous Dolynomial of deg ree n on A to E(A) is simp ly l a , wher e )- is of A and a is a fixed el eme nt of E(A). ·;;e are t nerefore l e:. t o consider f unctions which, i n the neigh- b or}-wod Vo_) p, are e xp res s ibl e a s a power seri es i n the form. "\ To s i mp1 if y the wor ::.: 1 et us make the as .;·wrp t ion t hat A0 = 0 , Wflich ar-:ount s si mply to writing).. for A - /\ 0 • f 8n 7 b e an infi n it e sequence of elements of 1 a co~plete ve c tor space E(A). Let r = l im 8n·l n be positive . Then Theorem ( 1~ . 41) Let I 11 - - ii co ,-- L )-. n ➔ oo n 8n converge s or diverges ac c ording l y a s ()/ is les ~- t h n or greater U' than r. Proof : 00 Let { ·,. / < r. In Theorem ( 14 . 262) re:place Xn by A n 8n . then ~ )._ n an converges absolutely, since t I)._ lj ll 00 of rand the well known Oo.uchy test, c onvergeJt. cited, the ref ore, 8n converges. In 8n is, by the definition By th e hheore~ already If \ /'- \ ) r, t l1en from the definition of r and the ordinary t h eor/ of p ower series t :r_e seqi..:.ence in~1\ ¾.I\! has n o Ul) ~er bound . Supuose 00 i) r), ~ ")..n a f or sor.ie v<>l ue of ). , ( / )- 0 were convergent . T~e n for arbitrary e ther e is r e s-:. :.ch tha t Sn I! . Sn - S )'. <: e for all n _> Ne. Let n be c ~10 sen > L..._ 0 ~ a Le t S be its sum . satisfies n Ne so tbat then (.= /!I sn+l - Sl1/ + il Sn - s 11 -<2e 1) This co ntradicti on proves the second part of the the orem. It follows fro m t c is theor em that a ny sequence < 8n~ of el ements 1 of a comp lete sp ace E such that r : lim l\ an \\n) 0 defines a function, ... ...._ c,Q 00 f()-) = ~ 'l ~ on (O)r of A to E wh ich is 0 by de f init i on regul& r r(O) at~ = 0. :.:oreover, Jranlf i s ?r e cisely rr~ o f t he mor e. g ener al de :Ci r. it orr so that the functi on f( )J define d by the s e quence ~-a Z . n . i s a l so 2nalytic r' (0 ) = r(O) at /. = 0 . Conversely: If f(1, ) o~ (O)r of A to E(A) is regular r( 0) Theorem ( 14 . 411) oo at \ = 0 a nd if f( \ ) = - ~ . ., nan _l in ( 0 )r (o) ' t h en r' = lim :· 8n·· n > 0. n ~oo 00 P roo :f : If r' = 0 , t h en t y T{1 eo r em ( 1 ~ . 41) ~ ) ._ n 8n is divergent for I- JO wi1ich cont r adicts thP- definition of regularity . Theorem ( 14. 412) I f in t h e above t heorem r( 0 ) > r' ) then f(A) is •analytic r'(0) = r' at A= 0. Proof: The added as sumption simply makes f ( ).) satisfy the condition r'(O) < r(O) of t he de finition of analytic functions. 00 T)-,eorem (1 4 ,92) I f f(A) = ~ / ' an is :"lytic r 1 (0) at ~ = 0 on (O)r'(o) of .A to E' and if f < r'(O), then f )~~ converges uni- forml y in /'- over ( 0) t> · Proof: In t he corollary of Th eorem (14 .263 ) take E0 to be (O) p and fn(x) to be )..n 8n· Since Then for ~ in (O) p we have L ? n \\ ~ )\ converg es th e conditi on s of the corollary are me t. Combini ng; thP- re sults of this theorem and The orem (1.; .261) we have the p roof of Theorem (14.420 ) of A to E', and If f()J is ana lytic r'(O) at 1 = 0 on (O)r'(o) if f z r'( O) then f( f-) is cont i nuous OYer (O) p . By Theo re:-- ( 14 . 264) we have: CO Theorem (14,422) I f f() J 00 ·= '> l'J ")_n ,~ an and g(:\ ) - ir -,.n bn are Then f( ~ ✓ + g()) is regul ar and is e oual to regular at 1 = 0. <D ~ ).,n(Sn + 'bzi). 1J co Theorem ( Hr .43) If f (.',) E ·o , .. n ~ is on (O)r(o) of A to E(A), regular (O)r(o) at / =0, and if for all /t in (O)r(o)' f(). ) 3 O, t hen 9n = 0 (n = O,l, ~,···) Proof: A 8 sume the contrary. Let~ be the first coefficient 00 which does not va nish, so that O = f( ) ~ = / - /'n 8n• Using Theorem (1 4 .261) we have for O -...: 1 > m 1 " r( 0) L '3. 0 _ - ~ , n-m soo /', r 8m+r = L- I' 8n = ). m f(A) rw-i 00 Therefore 1 0 • function of ,, at j = O. /\. _ r am+r is a re.RU].ar -- for )_ /; 0 is z ero. .And its value By T'·ieor e!:: (1 4 .421) itis co n tinuous; so th2 t its vc.lue 00 at :, = O must also be zero. · , r 8m+r 0 :But = 0 = 8tn =f O. This co ntradiction proves the t h eorem. If f() ,) is regular at 1 1 = 0 its expansion Theorem ( H .431) = O is unique. about \ 00 Proof: Let fC,, ) = 00 ;- )-..n ~ = 7 )__n bn. 0 0 Using Theorem ( 14. 264) and by the last theorem an = bn or Theorems .Qll Series of Polynomials. 14 .5 We sha ll now de :i uce a result wdch it was not co:: ven i ent to obtain in Division 13 because of the lac~ of ade qua te prelirr.inaries on co nvergence. Theorem (14.51) Let ~(x) be a s equence of homogeneous polynomials 00 on E to ~•, each of degree k. Let the series 00 i n some nei ghborhood (O)r. Then h( x) =L L 1 ~(x) converge unifo rmly hoW\.-0 9e.V1 eo-J..! hn(x) is a~polynomial of 0 d egree k. Proo f : over (0) . r Lt follows from Theorem (l~ .261) that h(x) is continuous Let hx,p(x) = ~ ~(x). It is the sum of a finite number of homobeneous polynomi e.ls of degree k and is t herefore one itsel f . The !!~a ximum of I( bn / x) \\ over ( O)r is IIU'lnp •z-k and over any other ( 0 )r is 1 d{ ~p•r • The condition for uniforni convergence is equiva lent to the cond ition that Tlli1up· rk tend to zero wit h 1 /n uniforml y in p. Since con- v e r g ence is as sumed uniforr:i over (O)r this condition is satisfiec_. r But 00 it is also satisfi ed for (O)~. (0)-. r By taking Hence bn(x) converges uniformly in r sufficiently large h(x) is proved continuous f9T every x. Uow b y the definition of a polynomial hxi(x + )_y) is a poly- norrtic-'l of degree kin f and by Theorem ( ,3. 204 the coefficients of ).. 8 in this polynomial can be expres s ed in the form k LA t=O ,; e st h(x+ty) have therefore that k 00 00 :[ hn(x + )-_ y) = 1 Ln LS=O k 'A s ~ Ast ~(x + ty) and u s ing (1 4 .264) we rearrang e tds sum so that k h( x + ). y) = L. s:0 r 6 k ~ Ast h(x + t y ) A t~o To s ~1 ow that h( x) is homogeneous we have The s - eming leng th of t h is latter portion of the argument may be exn l a i ned on the grounds that ~ompletenes s of E ' was not a s sumed . Theorem (14.52) Let l bn(x) be a sequence of homogeneous uoly:> nomials ea ch o f degree k on E to E' where E' is complete. Let ~ 1 mhn 00 converge. Then Z: bn(x) converges t c a homogeneous polynomial of degr e - 1 k w~~os e r:; od.ulus is at most equal oo to L mbn. 1 Proof : Theorem (1 ~ .263) establishes uniformity of convergence. h(x) = Therefore by (1 4 .51) above 00 'I: hn(x) is a homogeneous ,olynomial 1 of degre 2 k. Again mh = max 11 14.6 ~ h(x) )! ~ x l/ ~l 00 L 1 00 max hn(x) ! x ·. :~1 = (· -•·· ·r ~ ; Fjl.ndamental Prouerties of Analytic Functions In this section we shall de duc e theorems analogous to some of those in section (14.4). As regards the point x 0 around which the functions are i ~ the first instance assumed to be analytic, we observe ths.t there is no re 2.l los s of gener a lit ,y in taking it t o be the point 0. This amou-11ts si - ply to writing x in place of x - x 0 . Theorem (14.61) Let ~1\i(x)J be a sequence of homogeneous polyno~ials on E to ; • wh er e E' is c om~lete. 1 r 1 ( 0) lim (( IIU1nl/ -ii > O. Then = Let the degr ee of hn(x) be n, a nd let 00 L r1n(x) co nverg es and de i'ines a function 0 n ➔ OO analy tic r'(0) at x = 0. Proof : 00 Since - ·,r e have for :, x .< f <... r'(0) o n Till1n co nverges, -Vlr-- have by Thoerern (14 .263) that [ hn(x) 0 \ converges unifonnly and absolutely over (O) p. within o..._ Since pis aribtrary r'( O) the series converges at all points in (O)r'(o) and t he function it defines is regular. function is an&lytic. By the hypothesis on r'(0) the Theorem (14.611) and · if ~<. r 1 (0) then Proof: If f(x) = L,bn(x) is analytic r'( O) on E to E L ~(x) conve~ges uniformly and absolutely over Con:~,letene s s of the sp ace was not used. in ( 14.61) exce·pt to establish the esixtence of the limit. hypothesis 1 Therefor e , si nce by L I1i/x) converges to f(x), the argument of (14 .61) may be used. 00 Theorem ( 1 •4 . 61:?) a.re regular at x = · 0 ·t he n their sum is regular . Th eorem (14.264) gives Proof: But h.n(x) + ku(x) is homogeneous of degr8 ~ n. By induction it fol l ows that a s ~m ilar t h ing is true for a ny fi :1 i te number of functions. CD T'neorem (14 .613 ) If f(x) =L 0 h (x) is regul a r !t"' (0) at x=0 n and if f ' x) = 0 throTu~hout (O)tr(o)' then ~(x) = O. Pr oof: Let x be a fly point i n (O)t" (of ( 2.11 n) Then if\ >-.\ Z 1 By Theore~ (14 .43) therefore hu(x) = 0. Vve shall now c onsider the question of dif f erentiability of analytic functions. i1t this p oint it b ?Cor!;es nece s ary to draw distinctions be twe en theorems which ~1ol d for spaces E( C) but wh ich we do not s eem able to prove for spaces E(R). Theorem (14,62) If f(x) is analytic r 1 (0) at O on E(C) to E'(C), and if E'(C) is complete then the dif f erential fx(x; t x) exists at every point i nside (O)r'(o)· 00 = ~ ½i( x ). Choose a fixed noint x i n sid e · ( O)r' ( o) and select f ,f' so that x < f t.. f l. r 0). Let M = DPtf( 0). Pr oof: Let f(x) - I 0 1 0 ( I Equation ( 4 ) Theorem (1 3 .81) gives i n hn (x0 +LI x) - ¾ (x 0 ) - db,i(Xo A x) = If now Ax is restricted so thatf £1 x ~r(x 0 , A x) (1) n,r-• f , - \\ x \\have by the defi nition 3 0 of M that (2) Applying to 1'1n( x 0 + /J. x) result of the first pa rt of ine qu.ali ty (1) of Th.e or em ( 13. 741) we obtain (3) (4) and rc\~r Since knr(x 0 ;ih) is homogeneous in A x and since the inE.ql.nlity (3) holds for all!!. x in ( thence that O)p" it follows that II kfu,(xoAx) II ~ M (f' fl !I ~ kn,r(x ~x){\ ~ M{f-). n 11 A xlf fu lf' p"2 - f"l\A Xlf O' (5) (S) - N0 w s irn~ly sum the expression (1) fror.i Oto oo. The two series whose general tenns are L1n_ ( x 0 +ex) and hn( x) conver ge by the hypothe sis of the theorem and the choice of 6.x. The series L ~(x,llx) converge~ uniformly by (A) and Theorem (14.263) and t herefore by (14.51) is linear homogeneous. Denote by IIAx \l €.(Ax) the sum of the right hand side of ( l) Inequality (6) above gives that (7) . Writing down the sum of (1) ...Q.l f(x 0 + Ax) - f(x0 ) . (7) and the linearity of ~ - ~ dhn (x0 ;Ax) = \\ ~ x \j•e-<tix) i)o 00 !' ~(x ;&) prove that -i; ~(x ;Ax) is 0 0 the differential bf f(x) at(Xo)· Theorem (14.621) 0 6 If f(x) sa tisfies the conditions of Theorem (14.62), then the series l)1n(x)may be differentiated repeatedly term , by term. Proof: Since equation (4) (14.62) holds for every x0 in (O)p ~ '4 and since d.1\i(x;Ax) is of degree n-1, we have,"fheorem (l~,7?-3) r• f (ft ~ ~-1- pn-1 · • M = ,n 1 - ·l m(dhn)f n=r ~p r ·= lim Hence I n-..oo But f' is any number less than r'(O). Thus r ~ r'(O). · \', e c onclude that the differential L.dl\i(x) is analytic over (O)r'{o)· be differentiated again. exist. It can t herefore We assert by induction the.t all differentials Since the dif f erential of a homogeneous polynomial is symmetric in the i ncrements and since the onalytic function is the sum of such functions it follows tha t the _k th differential of an analytic function is syr.xnetric in the differentials. ...I" . The analogue of Theorem (14.62) for . spaces E(R) cannot be 4 p roven in the same manner. As a matter of fact it is still an open question whether in gene r cl it can be proven at all. However, we £hall prove. Theorem (14.63) If f (x) is analytic r 1 (0) at O on E(R) to E~(R) and E 1 (R) is complete, then the differential of f( x ) exists 1md. at any po i nt within (O)r"(or Proof: I t is given in tht:S region by Ldhu(x). 'Empl~y an argument s·i milar to that of (14.62). · Start~ng with equation (1) of that section: n ~ ~(x;/lx) we proceed by the use of f i(2l i r\ (1) f ~of bf 13, , 4 ) to . write the inequality (2) where h nr (x,!.x) is defined in ( l ~. C.1\-). As an upper bound for the ri ght hand side of (1) we have (?,) Let us now select f < r 11 (0), and a fixed :9oint x 0 in (O)p· so restricted that I( x 0 )) + /I A.x I/ <. p. Let Ax be With these restrictions the right ~nd side of ( 3 ) is a convergent seri es whose sum is less than a certain constant M rnulti -:_1lied by )\ Ii x jf. like that of ineorem (1 4 .62). From t his point on the argument goes ~rom equation (2) and Theorem (14 .263) the convergence of Z,. d.hn( x;,6 x) is established, and by ( 14 .51) it is :profed linear homogeneous in~ x. Thus f 00 I I ( Li f(x) - . (4) w~dch proves the existence of the derivative, for it is given by ~ L dhn(x;~ x). Furthe~nore, L 00 1 dhn(x;~x) is analytic; for we have, using l the results of (13,~2). (5) From t ·:is, (14 .61), and the hypothesis, we have therefore that L dhn(x;4 x) is an~lytic 'f'M( 0). Theorem (14.631) If f(x) satisfies the hypothests of Theorem 14.63), then it is differentiable of all orders, wit hin (0\ 11 (o)• ~: It is sufficient to show that the radius of absolute a nalyticity associated with a series is not greater than that of its derive d s eries. If t his is shown, then the theorem will follow by applying induction to (14.63). To prove the desired result we have from (\~.e2) that = n h(~, x, x, ••. , x) · and hence max n -h({lx,x1 ,x2 ,•••,xn-l) flx1Jfd ' 71 . . Now therefore we have n ➔ oo The left hand side of this equation is the radius of ab solut e analyticity 00 aasocia~·ed with L dbn (x; llx) . Applying what we h 2.ve just proved to the 0 (n-l)st derived seri es, wh ich we assume to have a radius of analyticity not le ss than r 11 (0), we p rove that the same thing is true for the nth derived series. Let us now turn our attention to· the question. of series wh ose genera l term i s 2 .n ana lytic function. 4- T},eorem (14 .64) Let f 1 (x),f2 (x), ... ,fi(x), •·· be a sequence of functions on E(A) to E'(A), v1here E '(A) is complete. Let_fi(x) be analytic r ! ( 0) at x = 0 and ltet R, t he lower bound of all r (0) be positive. 1 1 Let f be less than R 1and define M1 • converges, f(x) = ~ Dr fi (x). Then if M ~ L.. Mi Ll f 1 (x) is an an'. lytic function on E(A) to1 E' (.A.), Cl.) whose radius of an8iytici t y is not le ss than R. b:) Proof: 0 <. f 1 < f Write fi(x) = ~ 0 hin(x). and x r estri c ted to be in (0) fi. Let p be selected so that 1 Then by the definition of Mi (1) we have So that t he modulus of hin(x) is not greater than M1(p)n· From ( 14.52) we have immediately that bn(x) 00 = L h 1 (x) 1=1 n converges to a homogeneous polynomial of degree n wheee modulus is not greater than 7'2- From this it follows that (2) 00 and, by ( 1'"1 . 61 ), that L ( ~ x) defines an analytic function w;1ose 0 radius of an2lyticity is not less t han f· It is now desired to prove that f(x) = f hn(x). We have , rememl:ering always that \\x\\ < e_ , 1 (3) low 00 .·h erefore by (14.263 L f 1 (x) co nverges (unifo rmly) over (O)fl. lTow 1 let 6 be gi ven and select p so that Raving chosen p, select k so that (4) whi ch may b ~ done s i nce all the series i n (4) have been proved convergent. (5) But the second term on the right hand sid_e of (5) is less than or equal to and as for the third term of (5) i~ i~ riot greater than These results, when placed i n (5) show that /I f(x) - ~ hn(x) ~ I.. t Let us now ap ply the last theorem t o the__proof of: Theorem (14.65) If f(x) on ~C) to E 1 (0) E 1 (C) complete is = analytic r'(O) at x = 0 it is analytic at every point x in (O)r'(o)· ..22. Proof: Let f ( x) = ~ ~(x). Let x be chosen in (O)r'(o)· 0 0 We wish to show that there exists a se quence of polynomials Ka(x), deoo pending of course upon x 0 , such that f(x) ■ 1½i(x - x0 ) is analytic f at x = Xo and equal to f(x). and let F(y) = f(y + x 0 ) ·.:. o facilitate matter s let us write x - x 0 = y, = f(x), so that F(y) = f0 ~(y + x ) is analytic . 0 at y = -x 0 and has as its radius of analyticity r (0). 1 We shall orove ?(y) is ana l yti c at y = 0. Let p ,fi be selecte,~ such that O <- x 0 <. Let M = D f(O). f 1 <: p <. r' (0). It follows fro r;: the d efinition of M tb.at l 11n <x) < (1) 11 i Let fi (y) = hi (xo + y) = L hin(y) n=O (2) ce the usual re duction of a :polynomi a l in y to the sum ,: f h oi,ios eneous If y be restric t'? d to satisfy p olynomials. I' d Y 1jj' <"fz 11 f 1 - /l xc,~)I we •! nave from 'fb.eorere ( f3,7~ that (3) It follows that for all values of y (4) : row ap9ly Theorem (1,:;; .6L1 ) observing that f 1 (x) is a polynomi a l in y and therefore a particular instan ce of an anal:,rtic :tlu.:1ctio n, and t hat fro m , D ·,. i M (f l) is a f 2 dominarit of f 1(y). 'raking M1 in (14 . 64) e quation (4) f-1) \.f to be M 1 v.1e s · e that f Mi the theorem are met. Therefore, F(y) = l~ converges and t hat t he conditions of hn( y ), wh ere ½,(y) = ~ hi (y). 0 llow the modul us of hn(;y ) is s ee n by ap·'.Jlying (1 4 ,52) to (4) to be not greater Hence the radius ;,(o) of r' (0 ) ~ nalyticity of F(y) at y =. O satisfi es _l lim (mhu) n ~ lirn Finally, go back to the original variables x. Everything is t her efore pr eyed. This gives We note tha t by def i ning f suitably close to r'(O) we prove that r'(x 0 ) ~ r'(O) - (\x 0 \~ Theorem (14.66) Let G(y) be a h omogeneous polynomi a l of degre e k in Eb(C) to E11 (C), and let )J = f(x) be a function anal ytic r'(O) a t x = 0 on E0 (C) t o ~~(C). Then g(x);; G(y) is analytic r'(0) ~ ~'(0) on E(C) to E11 ( C). Select R, f M= M l - ~l f 1 < f < r'(0) and define M = D f f(0), • Th us for x in ( 0) f1 to satisfy 0 ( F n 0 w s ince G(y) is continuous and L~(x) co r.verges it foll ows, wri*ing lim G8 (x) = G(f(x))= g(x) (1) •➔ co Gs(x) is, by Theorem (13,9~, a polynomi a l in x of degr ~e ~ sk. Let W6 ( x) be the coefficient~ 8 in Gs(Ax) when it is expanded as the sum of homog eneous uolynomial s times powers of A• Let us first prove that if p) 0, G9 ("x) - Gs+p(1'x) contains no terms of degree ~ s in I' . Write G( y + z) in the f orrn where ,Jr(y,z) is of degree r in z nnd k-r in y. Y = H8 ( A >€), In t nis for r,ula write f -= ~s+l, JAz = Hs+p<Ax) - H °'x). Since the right side 8 of this last expres s ion cont ains no terms of degree~ s, z, as defined by it, will acti.ally be a polynomial in~- The coefficient J 0 (y,z) in (2) is, as has been verified, several times before, precisely G(y). Therefore we have G(y + rz) - G(y) (3) = G(Hs+p()_x)) - G ( H8 (Ax)) which is what we wanted. It follows from this fact that if r~ s, then the coefficient of )...r in G8 0-x) must be precisely Wr(x); for, if it were not, then G8 (~x) - Gr(r.x) would have a term of degre ~ r. Therefore, the first s+l terms in the expression of G8 (~x) as the sum of homogeneous polynomials j 71. 00 To prove that L W (x) converges to g(x). let S::O 6 so that b y what we have ju.st said Wr 5 (x) = W (x) when r ~ s. 8 Jow r estrict x to be in (O) p . 1 Using Theorem ( l~· 74) (4) from which it follows that for all x (5) and the r efore that if \ xl\ < f1 ( 6) Wr(x) converges to g(x ); for if e is a given nurr.ber we may take s so large that for s 1 >s , the right hand side of (6) is l es s than ej2 and tam select s1 so that ff G81 (x) - g(x)//<. e/2. Doing this we have {?) To se ~ that g(x) is analytic we have only to consider equation (5) which shows that the radius of analyticity must not be les s than ?i· S :nce may be as clo$:-e. as we like to r'(O) it follows that the radius of analyticit y of g(x) cannot be less than r 1 (0) . p1 78. Theorem (14.67) Let f(y) be analytic r 1 (0) at O on E 1 (C) to 0 E (C) where E11 (C) to E~(C). If there exist Rand G5 such that R = r 1 (0) and such thd 11 is complete. Let g (x) be a nalytic r'(O) at O on E (c) 0 for all x in (O)RI\ g(x)I\ < ti!)< r' (0) then F(x) ~ f(g(x)) is analytic R 1 ~ R at Select x = 0. p1 ,f f to satisfy O < 1 (. f < R. Define ~ (x) E J1( g(x)). preceding theorem K1(x) is an analytic function of x. By the Let us write We wish to sh ow now that K1(x) satisfi e s the requirements of Theorem (14.64). 'l o do so we replace G(y) in Theorem (14.66) by J (y). (5) of that theorem then becorr,es II IC:1p(x)I\ < mJ1 · 0! 1 Tnis s hows that Mi e mJi•~1 is a ~ ' r' ( 0), L M1 is co nvergent. l~r 1 Tnu s the series L K1 (x) of analytic where Hp(x) is the homogeneous polynomial, of degree p defined by Hp(x) a 2-: k1p(x) 1=<) - (1) f 1 dominant of K1(x). But since functions Ki(x) de L nes by T-heorem (14.64) an anal ytic function. CX) Equation We have 14.7 Analytic Functions fill1 Differentials in Composite Spaces. Although the re s ult s of the p resen t section have considerable interest in t hemselves, especially as a starting point for ful'the r work, t h ey are here directe·: almost entirely toward the differential e ..: uation p ro bl er.i whi c:1 wi ll b e discus 2ed in (14 .8). The manipula tions made :possible by the notion of composite spaces s eem to pre sent a very g lib way for proving theorems. ~ .::e shall., of course, make free use of the notation 3e.~+l~k introduced in,._(~ 2., tt-). Theorem {1 4 .71) Let Pi (x) (i = 1, 2, · · • ,k) be k p ol;m omi als, of degree ni on E to E1 respectively. Then p(x) = (p1(x),P2(x), ••• ,pk(x)) is a polynomial of degre e non E t o t E1, where n = ~ n1 • In particular if Pi (x) are all homogeneous of degree n, t 11en so is p(x). Proof: If x,x 1 are t wo points then \l p(x) - p(x') j\ is the . (I greatest of l\ Pi (x) - Pi(x') !· Let P1(x + 1'Y) Hence p(x) is continuous. n = 7 ~r k. (x,y), wh ere, of course, if the ~of\ 1r degre e of Pi is less than n s o~ e of the k's are zero. Then Hence p(x + ).,.y) is a polynomial of degre i? ~ n on AlE]to f.\E1 • Let (x,y) K l,-:.{ take on va lue s such t hat for s ome j kjn ~ O, then the coeff icient of ): in the last member of ( 1) i s different from the zero. is of degree eaactly n. o+ ctec,r-e.e l-'\ If p1(x) are homog eneous~we have Hence p(x + N Y) If p 1 ().) in the la st theorem are homogeneo us of Theorem (14 .711) degre s n, then mn = max mpi. i Proof: mp = max~ -n ( x)~ l x j\ x I = max I\ (I? 1 ( x). .• • nPk( x) J I /l xlJ . x = max X Theorer:1 (14.72) ma.x 1 /l ~ // = max if x,\ i (i = 1,2,··•,k) are analytic r (x ) I f f (x) 1 1 0 at x 0 on E to E1 respectively, then f(x) s (f1 (x),f2 (x),' •.• ,fk(x)} ie analytic r 1 ■ mtn r ! ( x ) on E t o i Proof: ·l lT ~ . O CX) Let f 1 (x) = ~ h 1n(x), For si mplicity take x0 = 0. and ho(x) • ( h 1n(x),··•,h1m(x) ) . Then by Theorem (14.71) hn(x) is homo- geneous of degree n and by (14.711) i t s modulus is the greatest of mhin· Hence -l lim (mhn) n = lim n~ n--,.oo Now let f be ~elected so that 0 ~ f < r'. \I · Then clearly ho(x) 1[ · <l.!\fI[ , r so that mhn ~ Mn• n lim ~ ( ~ ) n = lim n-)cx:) M ·, ~ '- J ,n ) mp Df f 1(o). From this we have _1 -l Let M E = f (2) S1nce P is at our dis~ osal t he inequality (2) proves in conjunct i on with _1 ( 1) tha t lim ll-;?00 (mhn) n = r' Bl That L~(x) converges to f(x) is evident since the difference Theorem ( h:07 3 ) Let h'x) be a homogeneous polyµomial of degree· n on E to E'; t i en the r th differential of h(x) considered as a function · of the composite variable (x,61_x, • ••, ArJ) on Er+l to E' is a lso a- homo- Its modulus is not greater than g eneous 9 olynorr;i::.l of degree n. n1 h ( n - r ) ! IIln--n Proof : \ie wish first L · • • ,~rx). Let us use the re~resentation of . {13.83 ) and write ~=.rove that drh(x) i s continuous i n the variable (x,.d 1 x, -ii e observe that dln(x) when expressed in terms of the nolar is the sum of ½er.Jr,JS of the type h ( BX + / \. X + J! •-. • + j x) in-s (2) :But the argunents ex + A. 1 x + · · · + J x are 1 n-s l continuous in the variable (x, J 1 x, · · ·, d rx) and drh(x) being the sum of where s is an integer. conti nuous functions of continuous functions is continuous. If in drh the argument (x,~ 1~,•••,I.Jrx) is replaced by 1 ( x +/Ju• [,· 1x' ' •·· ' u1 \ r x +~A ~1" 1x' ' •·· ' 1-., x +'\ A x•) the result . ' [ 1·1 x +r ~ r /\ LJ r h a polynomial in)-., since t he same t h ing is true for the expression ( 2). Therefore drh is a polynomial in (x,t1ix, · · •, £\rx). That it is h omogeneous of degr e e n may be prove n b y re placing A ix by }_L.1 ix and x by ix i n ( 1) . As for the modulus of drh we observe that . Theorem (14.74) If f(x) =f h:n(x) is analytic at x = 0 on E0 0 of E t o E 1 , then its r th differentfal · is analytic in (x,~x, · • • ,4x)on E Er 0 to i'. Le t us write X a (x,~x,, •• , J rx) and ~(X) ~: = drL1n(x). Then b •7 the la s t theorem I\i(X) is a poly,·omial of degr.ee n on Er+l to E' and is therefore in particular on E Er to E'. 0 last theorem not g reater than ( _J. less than lim n ➔oo (ffinbn) n n! ) 1 n - r = 1!m ( n ➔ oo • Since~ is by the the r adius of analyticit y is not . . n1 (n - r).1 _l ~11n) n = r"(O). Partial Differentials o. r~91t.)"' If f(X) = f(x1 ,x2 , .•• ,Xu) is a function on~(~1E1 ) to E' 0 and if on (Eic[~EJ) 0 to E1 , f is differentiable q~ ~ at X0 = (xi,xz, ... ,xk)o fxk<xo; ~ xk) is called the partial differential with respect to Xk:• ~ Aa The existence of fxk(X 0 ; ~xk) for each k does not necessarily imply the existence of fx(X 0 ; Ll i), the dif f erential of f with respect to the composite variable X. However, the converse is true; the existence of the differential fx(X0 ; 6 1) at a p oint X0 implies the existence of all 83 the partial di f ferentials f Theore~ (14.741) Xk at X • 0 This is shown in: Let H(X) be a homogeneous ~olynomial of degree k n in the •1ar iable X ~ (x1 , · · · ,Xn) on space vEi to E1 • Then H(X) is a k homogeneous polynomial of degre e n on any subs pace of TT Ei obtained by 1 equating to zero certain of the variables x. Proof: Let Y d enote a typical el ement of the subspace of Tifk Ei ob t ained by replacing by zero certain of the x'a, say ~ 1 , x_p+2 ,•·•,xk> so that Y: (x1 ,···,~) ■ (x1 ,••·,Xp,0,0,··•,0). Now . . k since Y is defined as a p oint o f a subs:pace of"G}Ei, H(Y) must be cont i nuous , H( Y + ~ Y0 ) must be a polynomial in 1' and _we must have H(A Y) = >- D.g(y). These thr ee propeities characterize H(Y) as a homo- ge neous_poly-l1om i al of degree n. The s~e procedure may ~e carried out f or any other selection of the subset of x's~ Let f(X) qe analytic r'(0) at I= O, where X is (. k \\" c o.,.,, p \ e +e an element (x1 ,x2 ,•••,xk) of(1;:_TE 1 (C\/o E 1 (C),l. then it is analytic on T~1eorem (14.75) any subset of TTE1 obt a ined by g iving certain of the x 1 s fixed values inside th ~ region of analyticity of X. Proof: xf<r'(O). Let x~1 ,···,~ be fixed values of ~ 1 ,•·•,Xn, whe re T"hen by The oren. (14,C..0 f(X) may be expanded about the point ( 0,0,•••,0, x~ ,••·,x!) in the form 1 If now ~ 1 ,•••,Xn take on ~he values ~ 1 ,··•,1, H(X) will by (14.741) · become a hom?geneous polynomial of degree n in (x1 ,•••,~). Furthermore, denoting by ~ the modulus of the pol~rnomial obtained by fixing these variables we have )'\ Theorem (14,76) If f(X) = f(XpX2, ••• ,Xn) on(Tf1G~}o. •1SJo . l'=-\ , to E' has a total differentialat X0 , then the partial differentials fXi exist and are given by where (AX)k= (O,O,··•,lxk,o, ... ,o). Proof: The condition for the ex istence of f 1 (Xo;AX) is that € (~x ) defined by (1) ( llA X )\ I 0) be continuous at ll X = O, and that fx(Xo;~) be linear continuous in tJ X. This must be true for all /l X and hence in particular for (Ax 1 ,o,o, · · · ,o). The cod.ntions a b ove then give that6 1 (~) defined by Gi_(~) 3 G (4 x1,0,0,•••,0) is equal to and is equal to O for A~ = O. But since fx in 6_x 1 , it must ~e the partial differential. 1 is linear and continuous Theorem (14.77) differentiable ~ x • 0 at y 0 = g(x0 ). Let g(x) be a function on E of E to E1 of E1 , . 0 0 Let f(y) be a function , in E~ to E11 differentiable Then F(x) = flg(x))is differentiable at x0 and the differe ntial is given by fg[g;gx(x 0 ;6x)] Proof: Define ' f 1 (ti Y) = \I l y I\ [ f( YO H ty< y Av)] (ti yf.o) Y) - f( Yo) - O; =0 t:/0) = II l x I\ [ g(xo + c:l x) - g{xo) - gx(xo ; tlx) E- 2< 0) = 0 f2(Ax) J (.dx/.0) so t hat by the definition of differentials € 1 (~y), G- ( &) are continuous 2 at iJ. Y = 0 and ~ x = 0, res3)ectively. l!ow = r(g( x 0 + ll x) - g( x 0 ) + y 0 ) = f ~ 0 + gx(x0 ;ti x) + = - f( yO ) / lox ' ' G"z(llx)) - f(y 0 ) (1) r~(y ;gx(x ;bx)) 0 0 1 + l 1 "- x 1'/· { ,iA.zlj · /I I ii ~ X 'I I where Observing the linearit y in ~ x of gx(x ;.1x) we see that 0 (A Y :\!) remains /J,... x 1 bounded ss Sx t en1s to zero as does the bracketed e~re ssion in equation (1). The refore the derivative of F(x) exists at x 0 and we have Theorem (14.78) If f(x) a (r1 (x), f 2 (x),··•,fn(x)) is the composite of n differentiable functions then ~roof: Let Lif.(x) = f.(x + t,x) - f.(x), and let 1 l l = /IA X/Jc' ~x) where 14.8 ! Differential Eauation Problem in Vector Space In this l e st s ecti on we shal l devote our attention to the solution of the differential equat ion :9roblem e ...actly; dy = f(x,y;Ax). To state t h e let f( x, y ,Ax) be an function of the variable triplet ( x, y, &c) analytic at (x 0 ,y 0 , 0 ) , on (EiE2 E1 )0 to E2 and linear in Ax. It i s Dos s ible to define a function y(x) analytic at x 0 on (E~ to E2 such 0 that y{x0 ) = y 0 and such t hat is satis fied identically in x,4x inside a sui t ably small ne~ ghborhood of ·-.:: the point (x0,y 0,Ax). In the discus sio ;: v:e shall write x 0 = y 0 = 0 simply to save writing x - x 0 and y - Yo• Un~il furtheF -~ent ion is made of the matter, we shal l consider t hat al l the spaces are spaces 1(0) .. In order to c enter t he discussion better, let us de rive a number of nece ssary conditions on the solutions of such e quations. Theorem ( 14.81) Ot\. If U( x,y) is analytic on ~ E ) to E', and -if y(x) 20 is a:-ialytic at x = o,/%\to E2 , and is such that y(O) = O, then U(x,ytx,) is analytic on! Ei\t o E'. Furthermore, its differential is g i ven by Ux [x, y; dX] + Uy [i, y; y~( x;eixil Proof: of Ei~· Consider the valuable composite variable X lly Theorem (14.72) the function X(x) x = 0 on(E0 0 to E1E2 • = (x,y) = [x,y(xU is ancdytic at Since y(O) = 0, it is always , os s ible to restrict x so that ljX{x)Q is suitably bounded and less than the radi us of analyticity of U(X) = u(x,y). This then satisfies the conditions of Theorem (14 .67). The differential with respect to x of U(X) is b y fheorem ( i 4-, 7 7) equal to (1) X = (x,y{x)] But since we have from Theoren (14.78) Ux [x,y;LlxJ + UY [x,y;yx(x;~x~, which completes Hence (1) becomes the proof. Theorem (14.82) to~ at (0,0;0). Let f(x,y;~x) be analytic in (x,y;Ax) on Ei_E;zE 1 Let it be linear in AX. Then if the r e exists a function y(x) anal ytic at x = 0 satisfying y(0) = 0 y X(x; AX) = f &,y ( X) ; Aaj identi cally in (x,4x) it rrro.st satisfy the condition that be syrn!Iletric in \ x, ~x.• By theorem (14.81) we have Proof: Since the le f t hand side is syin::.etric in 1 x, ~x the right hand side n;ust be also. As in the cas e of the classical theory it is, of course, not neces sary that the relation (1) above be satisfied identic 2l ly in x,y,~1x,¾x but only in (x,.Aix,.,1zx) wh en y has been replaced as a function of x. Let us now fix once and for all a function f(x,y;Ax) which satis f ies the restrictions ylaced on fin (14.82) and which in addition satis f ies the equation (1) above i dent ic al ly in x,y,~1 x,1x . The Symbol 4 Let U(i) denote a ny f u nction differentiable with respect t o~= (x,y). Let Ux(x,y;tix) + uy<x, y ;tiy) b e its dif f erential. ;·,e shall define the symbol 1 by means of the e ·uation By ~ 1u, o~U, etc. we sh&ll natur a lly mean the same function with bx G r ep lace d by ~ 1 x,~ x, •.• 2 If U depends up on oth er increments A 1x, they a re t o be treated a s constant. In terms of t i.1 is syrnbo 1 the sy...r!:etry '5::::ES~rt1iaam11 t condition in (1) of t he preced i ng theorem may be written Theorem (14.83) If U(x,y) is a function analytic in (x,y) ·at ' ; . (2t,y) = (0,0) on (EiEz\to E', t hen ~U(x,y) is a.na.lytic in (x,y) at (0,0) Proof: E1E2. Let X = (x,y) denote anoint of the composite space Write U(x,y) a U(X). By Theorem (14.74) Ux(X,AX) is ana l y tic in the composite pair (X,'1.X). 1~ow accor ding to the definition of ~U, it is obt a i ned sim"9ly b.' writing f(X;~ x) in lieu of A1 y in Ux(X,AX). I f we make t tii s replacement in LUC we have Now by (14 .72) the right hand member of (1) is, in view of the fact that f(X;t.x) is analytic and ~x i ·-- a const 0nt, ana l ytic in X. again, the variable pair Z ~ (X,41 1/ :By (14 .72) is analytic in X, so that ~ = flri,h:) Ux(X,'1X)I = ux< z) t1Y = f(i°'; A':t.} is a n a nal y tic f\:nction of an ana l y tic function Z. }J ZQ Furthermore s ince = ma.x ( 11 X I/ , ll lJ X // ) we can~ suitably bound the quantity I\X1and , through the linearity of JX in 4x, su.:.. tably bound l\ZU so that it al ways lies insid. e the radius of analyticity, of the function Ux(Z). Therefore we may app ly Theorem (14.67) to Ux[z(x)J and prove Ux(Z)_ is analytic in X. Theorem (14.84) If U(x,y) is analytic &n(EiEz) to E' at (x,y) = O, 0 Using the work of the preceding theorem, we have Proof: t:;y Let us denote by ~•X the function ob tained by replacing in~ by f(X,6x). Let us first calculate the differential of respe ct to X and an increment~· ~ix with We have (1) Rep lacing L~y b y f(X,~x), we obtain (Li!X)xG1;xJ = f6,fx(X,a 1x; 1xv (2) · = [o,D2 f(X),4ix] Now Taking t he differential of t r:. is wit h respect to X and an inc r ement Ll2X we have (3) ~he last seep is justified on the grounds that Ux(X;AtX) is lin ear in I\ its second argument and qua that argument is its own differential. Our result t hen becomes on sturing 4 x and using (2) 2 The first term is symm e t ric since it is the ordinary s ec ond dif f ere ntial of U and t he second is s yrrnnetric i,ecause of t h e hypotheses on f(X,41x). Theorem ( l •i , 85) If ~ 3 • • • ~ rep r e sent operators associ a ted 2 with- n i n crements .6 x, • • •, ~x then 1 1 s syrrmet ric in all the increment a A1x, ~ x, · · • ,-'\_x. 2 Proof: We prove this by i nduction on n. by hynothesis on f for n = 1. and apply Theorem (14 .85). The theorem is true As sume the t h eorem true for n-1. : ake There results But sin ce the left hand side is symmetric in Snon-1' so is the right; _for) by the induct ion hypothes is ~-l • • • Theorem (14 1 86) on{E1 < \ were commu t able. Let f(x,y,6x) be analytic in (x,yAx) at (O,O,O) EzEi}J) to a complete E2 (c) and linea r in x. i dentically in (x, y,Ax) ~ Then if we have · there exists a unique anal J~ic function y(x) satisfying identically in (x,l'.lx) the relation (1) ahd reducing to zero for x = 0. Proof: A repeated application of Theorem (l ~ .81) sh ows that if an analytic solution y(x ),' the eqmtions (l} exists, then its succes s ive derivatives rrro.st satisfy the relations Yx(x;t,_x) = f(x,y(x);Aix) y;x2(x; 6. 1 x, ~x) = O2 f(x, y; ~ x) Conversely if a function is a.na.lytic and satisfies the condition y(O) = 0 as well as the relation (2) then it is a solution of (1) since in par ticular it must satisfy the first equation (2). Now is a syrmnetric mul tilinear form in A x · • • ~ x since it is manifestly l ' ' n linea r in '1 nx, and is symmetric by '11heorem ( 14. 85). ~'i e shall prove that a solution of (1) i s given by y(x) = ~ ~(x) (3) where The hypothesis that f(x,y;6x) is analytic in the variable (x,y,~) as a unit gives us ea sily that f is expansible in the form 00 f( x,y ,Ax) - f(X;t..x) =· Z Hn(X,Ax) (4) n=l where X is the cowposi t e ( x, y) and wher e ~ (X,c6x) is a homogen eous polynomial of ,J. eg ree n-1 in X and linear in~ x. To s e e this, l et h.n( x , y,Ax) b e the h omog eneous :polynomial of degr e e n in (x,y,Ax) in the expansion off. Then is a homogen eous polynomia l of degree n in(~,~~ But since f and therefore is linear in Cix, the only term p ossible in this polynomial is that in whic:1. the degre e of p. is 1 and that of } is n-1. Therefore we have (5) ~y (13.93) bn(x,y,Ax) is a polynomial in X it is of degree (n-1). = (x,y) and by (5) Hence the HQ{;Ax) of (5) is given by ---- lT ow let ~-- be a dominant for the series (4) so that (6) Let us define~ ~(X) for i> 1 as the form of degree kin the expansion qi.J(x) of b1 f about X = O, and for the purposes of synu;:etry def in e Jf(x) a s t he' te rm of de gre e kin the expansion (4) , so that ~(X) = Hic+ (X) 1 we have and (7) where6i+lX is as defined previou~ly the composite function We may write A• X in the form of an analytic expansion as i+l ~sing (8) we proceed to pick out the terms of degree k in (7). We obtain Srt must be r ememberect of course tha t the expression L(X) involves Ax as well as X. Let us make the followi ng observat ion on the differential of "ti e have b ~,: (1~.g2) a h omog en eous :9olynomial of de g-ree r. (10) In what fo l lows we shal l asswne that t h e increments A1x,~x,•••,4ix,••• are fixed and thati\ 6 ix \ji s b ound ed, say )( 6 ix l\ 4' w It will t e understood that the r:·odulus mr{ s t and s for t he modulus quA X. 'i.' e s'.:all write~=~ for(i !c l; k > 0 ~and for (i) l; k ~ 0) and shall define A1 as t ne g r eater of~ and ma:: l\Lo ll OJ o l~\tl We now obtain from (9) ( 11) which g ives fu k+l ~+l ;;, l i e r ~ Air-r+l (12) ·,:e now mc.ke t he f ollowing observa t ion: b e :9osi ti v e numbe rs sc Lctec. a t r c;.ndom so t h e. t ~ g 1 l e t ~ (k = 0,1,2, ... ) Ai. Le t ~ ( i > 1) b e de f i ned L c.uctivel y by (1 2 ) with it s inequalit y sign r e --,lace d by equalit y . The .1 ~ ~ Ai: all ( i,k). f or, i f v; e 11ave up t o so:::e i, 2. i1is is ea s ily prove c. b.:i: i nduction ; 1 ~ ~ ~, then co Co n sider nov, a set o .:~ p ower series• 11 ( z) = T E; zr *Their convergence do es n o t co ~cern us. ~ The modified expression (12) the n g i ve s t hat li+l(z) is e ti rr: es t he Cauchy produc t of ¢1 ( z ) and the derived series of ~i ( z ); i.e., )Tow put ~ = M bk where fJ is the g reat er of A~ and M of equation ( .;. ) and where b = 1/p, have being that of e quation (4). r; i tb. t h is cho ice we -1 ¢1 (z) = (1 - bz) M ¢2( z ) = ( 1 - b z ) -3 ( Mb e ) •M From t h is we obtain in particular B~ = 1· 3 •5 • •.•( 2n-1) • (Mbe )~ n<. 2•4•6· • •2n · (11b e) M - ) n-M = n ! (2Mbe :-row g oing back to tne original se t up) the expr e s s ions Onf( 0) = L~( 0) are multilinear in ~ 1 x,4~,···,f+ix and wit hU~ ix \j(W we have of Therefore the modulus,.h.n(x) of equat ion ( 3 ) s a tisfies and w lim )0 2Mb e n ➔ OO 00 Therefore the s eries ; ½i(x) L1 (3) de fines an analytic f"W'lction y(x) such that y(O) = 0 and such tha t its dif f erentials satisfy the sequence of relations (2) at the point x = 0. The function g(x;.1x) = Yx(x; 6 x ) - f (x,y(x);tx) is an analytic function of x wnich together with 8.11 its differentials vanishes at X=O. 00 Since i n any anal ytic function F(x) = L Hn_(x), the differentials dnF(X) 0 satisf~ the rela t ion the f unction g(x;~x) must be ident i c&ll;'.l zero. is an analytic solution. Therefore y(x) = 'f_0 h n (x) I t is furt nerr;ior~: uni q_ue since the calculation of r1u(x) i s uniquely indicat ed. by the se quence of relations (2), together wit1.1 the condition y( O) = 0. Th e theorem is t herefore proved . . It may b e noted i n concl ud.ing ti1is section that t he formal steps of Theo~em (14 . 86) and the inequalit ies used are independent of the o.te fact that the s-pa.ce s~E(C ) , except in so f a r as Theorem (1 4 . 67 ) was used to justify the manner of obt aining (9). e bel ieve t hat the analogue of (14 .67) may be provea for s~aces E(R) at least inside a sui tab l y restricted regi on, but we have not yet succeeded in doing so. Part 2. VARIOUS THEOID:.~S ON THE REPRESEt'l""TATIOU OF FUNCTIO:JAL FORMS AND TRA:t;r8FORMAr ·IONS BY 1:EA.Ns OF STIELTJES INTEGRALS 21. Introduction to Part Two. In this second part we shall give a proof of t he representability by means of multiple Sti eltjes integrals of the most general multilinear form on the space of continuous f unctions to the space of real numbers. We shall also g iv e a proof of the generalization to Ri emann- Stieltjes i nt egr a ls of Arzela 1 s t h eorem on necessary and sufficient conQitions for t enn by te rm int egr acil ity of a convergent sequence of Riemann integrable f unctions. Tne proofs given ar e b J el em entary methods; i.e., wit hout the -J.Se of any but the mor e el ementary theorems of point s et theory. We shall ass ume familiarit y with the definition of and the more or less standard propertie s of s i rrrple Riemann - Stieltjes integrals and of functions of limited variation in a single variable. Also we shal l assume the funda- r-:·. ent al t heorem of F'r ederic Riesz on the representation of linea r conti nuous functions by means of simple Riemann- Stieltjes integrals. The majority of the assume d theorems and definitions are s tated in section 22. 99 22. Preliminaries .QB. Notation and Results Assumed 22.1. Subdivisions of!. linear interval: Let (a,b) denote a fixed closed interval of the real x-axis. By a subdivision of (a,b) we shall understand division of (a,b) into a finite number, n, of closed intervals (xi, x + ), where (i = O,l,··•,n-1), 1 1 and where a= xo< x1 ( Such a subdivision we shall designate •·•<xn = b. by a single symbol A. In general, ~,tl' ,A",•••, etc. will denote distinct subdivisions of (a,b). Two subdivisions A and A' are said to be equal when they have identical points of division; this equality is expressed by A== ~'. If every point of di vision of a subdivision A is also a point of division of a second subdivision A', then~ is said to be included in . A' and the fact is expressed by the formula ~ f ~• . If .6 ~ 4' and if .4' •contains a point of division which is not a point of division of A, then .6 is said to be properly included in A' and the fact is expressed by /lC /:J'. If /l and ,!) 1 are subdivisions and '1 11 is a subdivision which has as its points of division all the points of division ofAand 4 1 , and those alone, A 11 is called the superposition or sum of A and 6 1 ; we denote the superposition by ti 11 = A+ A'. By the norm or modulus of a subdivision J!,.we mean the maximum of the lengths of its intervals. is used to designate the norm of~. The syrnbolll~II The necessary properties of these relationships among subdivisions of (a,b) are given in the following: Theorem ( 22. 1l): Subdivisions of ( a, b) being denoted by 4, A', A", the following relations hold: (I) If l& g 6 1 and 6 1 ~ ~ 11 then Ll~t;.'1 (II) If ~ c A' and .A' ~ A" or if ~ ~ .1 1 and~• c A 11 then Ac 6" (III) ti+ '1 1 = A I + .A (IV) (LI+ A') +L1 11 = '1 + (Ll' + .1 11 ) ( V) If '1 u = l1 + A' then A ~ c1 11 and .4 1 ~ -.1 11 (VI) If fl~ '1 1 then II.All ~ Ii i! 'If ( VII ) If fl + ~' II ~ /I411 ; II A + A ' II ~ t1 Ll I Pfoof: definitions. 'f; Properties (I),•·•,(V) are itr1Dediate consequences of the To show property (VI) it is sufficient to obsetve that every interval of~• lies inside or coincides with some interval of /j. Property (VII) is a consequence of properties (V) and (VI). 22.2. Subdivisions of~ Rectangular Hyperpa.rallelopiped. Consider a rectangular cartesian hyperspace of n dimensions whose points are given by coordinates xl, ... ,xn. Let Rn be a fixed rectangular hyperparallelopiped whose projection on the x 1-axis is the closed interval (ai,bi) (ai bi; i = 1,2,•·•,n). In brief, Hn is the . set of points whose coordinates satisfy the inequalities ai ~xi~ bi. Now for each value of i (i = 1,2,•·•,n) let Ai designate a subdivision of the linear interval (a1 ,b 1 ) by points x~,·· ·,x!1 , where ai = x~~xi, < · • • < x~ = b 1 . ~ is by this process divided into closed hyperrectangular cells mix m2 x ··• x ~ in number, a given cell being made up of those points whose coordinates satisfy the inequalities x} 1 ~xi~ xji+l' (i = 1,2,••·p; J1 = 1,2,•·•,mi)• Su.ch a subdivision of Hii we shall designate either by a single symbol A or byd 1 -~2 ••• dn, where A1, a2 ,··•,~n produce A. are the subdivisions of (a1 ,b 1 ),·••(an,bn) which The equ.i valence of the symbolisms will be expressed by writing Ll !! LJ 1 .1 2 ••• }.n. Two subdivisions ~ !! fl 1.62 ..• An and lbO ~ 1 • ~ 1 !A 2 ' •.• ~n• of Rn are said to be equal if lli = ti' = 1,2,•••n). A= f,1.,J2 .. •/jn is said to be included in '1' a ~1•42' • • ~• A subdivision it J i ~ t:1 1 ' (1 ( i = 1,2, • · • ,n). Th is fact is denoted by ti & L1 '. tJ.. is said C •t to be properly included in~• if at least one of the inclusions~ i = ~1 is proper; in t h is case we shall write Llc '1 1 • two subdivisions Ji • ~ 1 . J 2 ••• A.n, 21 I The sum or superposition Ll" A'= '1 11..1 2 ' •··An' , ".f" !! (.1 1 + ~l )( A2 +~r )··•{L1n + An') if 8n is defined by and is written 6" = ..1+.L1'. Finally the modulus or norm 11~1 of A s t1 1tP of Rn is defined as the greatest 4f ti A' il , •••,JI tin!/ . We have, relative to these relationships Theorem (22._21): are subdivisions of Hn, then the following properties hold: {l) If L1~ A' and J ' ~ ~ u then A g L1" (II) If ;j c A' and Ll ' ~ Ll II or 4 i L\• and L] ' c A II then Ll c "'1 11 • (III) .1 + j I = LJ' + /j (IV) ( fj + 4 ') + Ll " = L\ + ( 1} + ti.") ( V) If ti 11 = L:i + Ll ' then L1 g A II and A ' § L1 • (VI) If A ~ A' then . lli1 II ~ II~ ' /l (VII) II 4 + Li 'II ~ II D /I ; II A + l ' ll ~ II A' fl Proof: Each of the properties is an i mmediate consequence of the definitions and of the corresponding property sb own for subdivisions of linear intervals in Theorem (22.11). 22.3. Differences of Functions wi th Respect to Subdivisions. Let !(x) be any real function of the real variable for a~ x ~b; let~ be an arbitrary subdivision of (a,b). a = x 0 < x 1 c:C • • • ~ Xui = b be the points of di vision of Ll. x def i ned Let We shall define [j1/, the difference of , with respect to the interval (xi,xi+l) of A, by • Let ~(x1,··•,xn) be a real function of x1,•·•,xn defined over a hyperparallelopiped Hn· Let h. a L1 1J 2 • • •/J.n be a subdivision of Rn· the notation of (22.2) we shall refer to the cell of on the x1-e..xis is the interval Using whose projection (xj , xt+ ) · ( i = 1, 2, • • •, n; J 1 = 0, 1, · • •,m1_1) 1 1 We shall define the difference, /\..i I -J1J2, • · ·Jn ' of </(xl, • • • ,xn) with respect to the cell (jl'j , • • • ,jn) by means of the 2 formula as the ce_ll (j 1 ,J 2 ,•·•,jn). where the symbol ~3i is regarded as an instance of t~e operator ~j defined in the immediately preceding paragraph, and is interpreted as operating upon all that fo H ows it quA function of x 1 • For example, if n = 2, we have 1:, Jiji s 1it ~ ;/(xl,x2il LlJljJ(x'.l. ,xjz+l - d(xl ,xj2;J = = !(xJ1+l'x;z+l) - !(xJl,x~z+l) - !(x31+1•:EJ2) + J(x}l,~2). 22.4. Sets of Linear Intervals Consider again the closed linear interval (a,b). 12 ,•••,Ik denote k closed intervals contained in (a,b). Let 11 , We shall employ the cocrnon usage of point-set theory in def ining J, the sum of I 1 ,I 2 , •• \Ik, as the set of points contained in at least one of the intervals I 1,•••,Ir It is shown as one of the elemenpary theorems of point-set theory that such a set J is made up of a finite number of closed, non-overlapping sub-intervals of (a,b). It is convenient to write J = I 1 + I 2 + •·• + Ik and the theorem cited states that there always exists a second set, Ii,••·,1rii of non-overlapping closed intervals such that J = I 1 +·•·+I~. Io 2.. Let J be the sum of the non-overlapping closed intervals I 1 ,r 2 ,·•·,Izn, and let A be a subdivision of (a,b). If each end point of the intervals I is also a point of division of L) then we shall say that J is included byL'.) and write Jcb. It is clear that if Jc.A then J may be regarded as the sum of those intervals (xi, xi+l) of~ which are contained in J. By the sum J + J 1 of two such sets of intervals we shall mean, as is usual, the set of points occurring in either J or J 1 • The sum of two J's can be shown to be a J. 22.5. Functions of Ljmited Variation. We shall now recall briefly the definition of a function of limited total variation and for convenience state some of the standard theorems on such functions to which it will be necessary to refer. Let /(x) be defined on the interval (a,b). of non-overlapping closed intervals I 1,••·,1m· of (a, b) such that Jc .Ll • Let J be a set Let ~be a subdivision . Then the number VJ~ I defined by where the surrmation extends over all i and such that the interval (s1 ,x1+ ) 1 of~ lies in J, is called the variation of </ over ~ With respect to /1. If there exists a positive number M such that for alll1for which J 4:; ~ we have VJ-f /< M, then / is said to be of limited total variation over J. The lower bound of numbers M having this property is called the _total variation of I over J and is denoted by VJI• In these definitions the set J may in particular be taken to be the interval (a,b) itself. In t his event we denote the variation of \ r/ over (a, b) with respect to .6 by V(a, b) ,r/1 and the total variation of , over (a, b), if it exists, by 'ca, bl· - The folloWing properties of functions of limited variation are standard. Theorem (22.51) If r/(x) is of limited variation on (a,b), then there exist two bounded positive monotone increasing functions p(x) and n(x) such that 9(x) = p(x) - n(x). Theorem (22.52) If 1(x) and ,/Ax) are of limited variation on (a,b) and c is a constant, then the functions ¢(x) + tJ..x), l(x)•!(x) and c•r/(x) are of limited variation and we have moreover V(a, b) ( ~ + </JJ ~ V( a, b) r/ + V(a, b V(a, b) c • )lf' f ~ (ma,:c i) ·~a, b) Y, + (rna}(c ?jl ) •V(a, b) ¢ Theorem (22.53) If ~(x) is . of limited variation over (a,b) and if J is a finite set of closed non-overlapping sub-intervals of (a,b), then r/ is of limited variation over J and V(a,b)r/ ~ vJ, Theorem (22.54) If J and J' are two sets of closed non-overlapping sub-intervals of (a,b) and are mutually exclusive except possibly for certain of the end poi nts of their respective intervals and if /(x) is of limited variation on J and on J', then l(x) is of limited variation on J + J• and l 04-_ Theorem (22.55) If ;(x) is of limited variation on (a,b), then (a<:: x ~ b) at each point x x (a~ x.t:::b) </(x + 0) ¢(x - ~) =h~+o lim r/(x - h) and at each point = lim ~(x + 0) exist. h+-+0 Definition. If r/,(x) is of limited variation on (a,b) and If, (1) ~(a) = o, and (2) r/,(x) = l(x + 0) (aL:: x~b), then ,ex) is said to be regular. 22.6. Simple Riemann-Stielt.jes .Integrals and Their Properties In tenns of our notation the definition of the integral 1 f(x) d/(x) may be stated as follows: Let t(x) and r/,(x) be defined over (a,b). If there exists a number I such that to 6 >0 corresponds ~/). 0 such that for any subdivision LI •atisfying llb//<oe we have where } 1 is any point in the interval (xi, si+l) of D, and where the sum ::ation extendes over all intervals of b, then the number I is called the Riemann-Stieltjes integral of f(x) with re·s pect to !(x) and is de- t(x) d~(x). noted by / a The following theorems conceming Riemann-Stieltjes integrals are well known and we shall not indicate the proofs. Theorem (?2,61) If /f(s)d~(x) exists and if c . -point of (a, b), then t aflf(x)d~(x) and Ja C is an interior f(x)d/(x) exist and A9 a corollary to Theorem (22,61) it follows that if l f(x)dl(x) exis_ts the~f(x)d~(x) exists when c, d, are the end points of :ny closed C interval inside (a,b). Theorem (22.62) If t(x) and ~ (x) are both integrable over (a,b) with respect to¢, and if c 1 and c 2 are real constants then c 1 f(x) + c 2t (x) is integrable with respect to; and we have Theorem (22.63) If f(x) is integrable over (a,b) both with respect to ~(x) and with respect to X(x) and we have Theorem (22.64) If/f(x)d/(x) exists then /b¢(x)dfCx) exists and a · /a T)1e2rem (22,66) Itj_bf(x)d~(x) exists, if ''(x), the derivative of ~(x) exists, and ifibf(x);'(x)dx exists as a Riemann integral, then a .. I o C.- Theorem (22.66) If f(x) is continuous and if t(x) is of limited variation on (a,b) then rf(x)dl(x) exists. a By the maxirrrum oscillation ~ f of a function f(x) with respect to a sub-division~ we mean the greatest of the numbers (f1 - J ) where 1 1 1 denotes the upper boi.m.d of f(x) in the interval (x1,Xi+i) andJ'i the lower bound in the same interval. Theorem (22.67) If ~(x) is of limited variation V(a,b)~ and if rf(x)d9(x) exists then a Theorem (22.68) If ~(x) is of limited variation V(a,b)¢ and if }t(x)d9(x) exists then a Theorem {22.69) If ¢(x) is a regular function of limited variation then the upper bound over all continuous f(x) of the expression Jf(x)dl(x) b a _ _ __ is equal to V(a,b)¢ Theorem (22.610) If ¢(x) is of limited variation on (a,b) and fn(x) is a uniformly convergent sequence of continuous functions tren 22.7,. ~ Riesz Theorem .Q.!!. Linear Continuous Functionals In tenns of definitions and notations of part I the definition of a linear-homogeneous continuous functional of a fun ction continuous on (a,b) is silll!)ly that of a homogeneous polynomial p(y) of degree one on F( a, b.) to R, where F( a, b) denotes the space of real continuous functions y on ( a, b), !IYII being defined as ma~ IYI . In the present discussion it seems convenient to employ the notation of the classical theory of functionals. The general linear-homogeneous continuous functional of a continuous function y( t) ,( a ~ t ~ b) we shall denote by L [y( t )] , where t is, so to speak, a dummy. The following important result is due to Frederic Riesz•. · Theorem (22.71) Let L[y(til be a linear homogeneous continuous functiona l of a cont inuou.s function y{ t) function of t (a ~ t ~ b). depending upon two parameters n, Let '7 n ( t; -c) be a (n = 1,2,3, . .. ;a~ T ~ b) and defined by "?n(t;'t) - 1 n(t - i) rn(t;i-) - 1 '{n(t;t) =0 (a = ?: + l /n = t = b) •• then the following conclusions hold. I. For ( a~ z-~ b ) lim L[rzn(t; t')] exists. n +OO II. The function~(~) defined by <'{' {1') = lim n-+oo Lf1ln(t;t)] a.::: r ~ b <t(a) E 0 •F. Riesz--Ann. Scientifique de l'Ecole. Norm. Sup. Ser 3, Tome 28, p 33, 1911 Ibid, Tome 31, p. 9, 1914. **It is unnecessary to define Y?n( t;~) for the case "'t'~ b "' 1:: + 1/n, since for fixed t and sufficiently .large n this inequality cannot hold. is of limited variation on (a,b). III. For any continuous y(t) r y( t)dr:f( t) = L[y( t a IV. v( a,b )'\' = max Y I ~ L~jt)J 1, where 1/y// = max y(t) IY a ~ t ~ b As a corollary we may show that the function ~(t) defined in II. is a regular function of limited ve~iation. . ct ( t) We have by II. and III. for r = lim L ['ln( t; r)] = lim t?n( t; t) cl.Q'( t) n +co n ~ooa (1) Using 'I"heorems (22.65), (2. . , .61)., (22.65), we have (2) where we have put h = 1/n. Now since~(t) is of limited variation it follows that ~( t + 0) exists and from the theory of ordinary Riemann integrals that the limit as h+O of the last expression in (2) is precisely C(( t + 0). Hence we have that for (a.lo t"<b) ct( t) = <!( t+ 0). and by the definition ~(a)= 0 so that ~(t) is regular. 22.8. Functions of Limited l,~ultilinear Variation. Let ~(x1 ., .•. • ~,xn) be defined over a hyperparallelopiped Hzi, and let /j !! 1? •t? •••/Jn be a subdivision of Hn. Let the number VH n, b ,I be Joe defined by where the slIDll'D8tion ext ends over all cells of '1 , and where the numbers 1 n Ji Jn ~ . , • • · ,€. are taken to be 1 or -1 in su.c_h a way as to make the sum as large as possible. ~ vli.a, 6 ~ is called the multilinear variation of¢ over with respect to A. If there exists a number M,>O such that for all subdivisions /J we have M>VH ~;, then I is said to be of limited multilinear variation. , n' The lower bound of numbers M having this property is c ~lled the multilinear variation of; over~ and is denoted by V~~The properties of funct i ons of limited multilinear variation which we shall have occasion to use will for the most part come as by products in the argument. A function! of lirr.ited multilinear variation is called regular* if it satisfied the conditions (i = 1,2, · · • ,n) except upon the boundaries of Rn (2) ;(al,x2, ..• z:ll) = ¢(xl,a2, •.. ,xn) = 22.9. Definition of Multiple Riemapn-Stielt,jes Integrals Let f(x1 ,·••,xn) and ~(x 1 ,···,xn) be 4efined over~- If there exists a number I such that to every E' > 0 corresponds aE> 0 such that. for any subdivision D. of Hn, satisfying IIAII < ~ •This definition is due to C. A. Fischer. we have Jl 0 ,.n 1 ,-2 . where (tj ,c,j ,·••,~j ) is any point in the cell (J 1 ,J ,••·,Jn) of.A, 2 1 2 n and where the surnnation extends over all cells of Ll, then the number I is called the multiple Riemann-Stieltjes integral of (°l bn f with respect to ~over~ and is denoted by) _, ,··•, ( f(x 1 ,•••,x?)d¢{x1 , ••• ,xn). a1 Jan 1 1 The limits a ,b , are of course simply the end points of the projection of Hn upon the xi-axis. Since we shall be interested only in a special case of the multiple Riemann-Stieltjes integral there seems to be no point in stating its more general properties. I I1 · 23. Representation of Numerical Valued Functional Polynomials ]?z Means of Multiple Stieltjes Integrals. 23.l. Outline of the Problem. We wish to give a. representation as a sum of certain multiple Stieltjes integrals on F(a,b) to R, where r(a,b) is the space of real functions)(x) continuous on the linear range (a,b) and where /I YII is defi ned as ~4~(x)I. The results on abstract polynomials show the representability of every such polynomial as the sum of homogeneous polynomials. Theorem (13,4'lr}. The intimate _connection shown in (,:;.t..2-3) between a homogeneous polynomial of degree n and its nth polar, whi ch is a mu.ltilinear form, brings the problem of finding a representation for a polynomial down to the problem of finding a representation for a The representation for the case n = 2 was given by b bl Frechet•, who showed that a bilinear form B[y 1 (a), v (t)] could be re2 al a multilinear form. presented by an integral in the formula where the function u(s,t) is of limited multilin ear variation. A new proof of t h is result and a remark as to how the proof for gen eral be ca.1·rieu out were given by C• .A.. Fischer••. n could ·.:.e shall give a more or less independent proof starting from first principles. We do this for completeness and for the purpose ot deriving some relationsnips we shall •M. Frechet, Trans. Am. Ma. Soc., V. 16, 1915 , p. 215. ••c. A. Fischer, Proc. National Acad. of Sciences, V. III, 1917, p. 640. \12 need in the characterization of the general polynomial on F(a,b) to F(a,d). As in the statement of Riesz' theorem it seems convenient to employ the notation of the more classical theory of £unctionals. 23.2. Representation of Numerical Valued Multilinear Functional Forms h means of Multiple StieltJes Integrals. Theorem (23.21) Let y 1 • y1 t 1 (a1 ~ t ~ b J i = 1,2,•·•,n) be n contin- uous functions on the indi cated ranges. Let W~ W(y 1,•··,1n) be a multi- linear functional fonn in the arguments y. Then there exists a function . ~ u(t 1 ,• , ·,t n ), defined for (a 1· = t 1· ~ b i ), such that for. all continuous y · the integral exists and is equal to W. Proof: Let M be the modulus of W so that De f ine n continuous functions~- 1,m (i = 1,2,•··,n) oft depending on a continuous parameter m. ( t; t ) and an integer By means of the formulas = 1 - m(t - ?: ) (a1 ~ t < t ~ t'+ ljm ~ bi) E O Put W(m; -rj (ai ~ TL i- + 1 /m L t ~ bi) = W(ml' • • • ,~; -z-1, •••-z:ll) ;;; W[ 1?1 ,m1 ( tl :1:1)' ••• ' '; n,Inu ( tn; '/;nil We shall now prove by induction on lim lim n that ••• lim W(m,zj m -,..oo n exists and that . defined by u =u(,?-, ••• ~) u( --r:1, · • •,,l) = lim ••• lim ml--J-00 W(m;_i-) (aiL: -c- 1 ~ bt) Tn.if<D (2) u(al,--c--2,··•,?) = u(f,a2, ••• ,0n) = ..• = u(~l, .•• ,tn-1,an) • 0 (3) n ~ M L/IY1ll •·• IJ Y1-1 ll ·IIY1+1jl .. ~IJ ynll ~iYi i=l where ~iYi is the oscillation of y 1 with respect to the simple subdivision Ai, and where (t~ ,···,~~ -)is a point in the cell (j •••j ) of • 1 Jl Jn n We first observe that _(2) and (3) are, for the case n = 1, true by virtue of Riesz 1 theorem. u(-cl) = For n = 1, equation (2) becomes lim ml~ oo u(a1 ) = 0 and the existence of the limit is proven by Theorem (22.7). •-becomes •which is true by (22.7) and (22.67) !~equality (3) fl 4 A Now I qua functional of Yn, is linear and continuous. From the Ri esz theorem (22.7) we have that v = v(y1 ,•··,Yn-l•r) defined by v=O exists, that (4) and that (5) »nploying Theorem (22.67) and equations (4) and (5) above we obtain (6) Now, for each value of~' v § v(y1 ,••·,Yn-l;t°) is a multilinear form in y 1 ,···,Yn-li for, if c and c' are constants, we have = lim 1f ~yl + c•y • ,Y2 , • · •,yn-1; '7 1 m -+oo n,m 1 = lim {c W [yl' .. ',Yn-l•'?n,J + c 1 W[!]_,···,Yn-l,'~1 ,m7} m-.co • (7) and similarly for the other arguments f• Also~ (8) Having proven by equations(?) and (8) that vis a IDlilltilinear functional in (n - 1) arguments we make use of the inducti&n hypothesis. Writing out the equation for v analogous to (2) for Wand assuming the induction correct for (n - 1), we have the existence of :But from the definition of v, this last expresse, tp.e iterated limit (2). Thus the existence of u is established. We now proceed to prove the inequality (3). Let ! = {an,bn) L Yn(c jnn )j j n v, which is simply the surr. occurring in the left hand side of (6). Now! is a linear combination of terms of the type v(y1 , .• •,Yn;tj) and is hence a multilinear form in Y1,···,Yn-l• Moreover, fror!" the equation (4) we have that fi A J ~~It~ /yn(C'j ) /·V v 5 M //11 /1 • • •I/YJI Jn n (a"b 11 ) (9) n 4 Hence a qua functional of y 1 , •• ·,Yn-l is of modulus les sthan or at most equal to M1/'iYn/J • Assuming now the indu~tioh hypothesis we write an inequality ana logous to {3 ) in which we replace W by~ and ~ e d 1 ••. Jl by the division .,., 1 •• • /Jn-1 defined over the hyperparallelopiped Rxi-l whose points ~ (t+, •.. ,tn-l) satisfy the inequalities a 1 ~ t 1 ~bi (i = l,2,·•·,{n-1)). This gives , 1 , 2 ·•· , n-1) and when sa 0 when for any i<- n -r;1 • a 1 . Replacing a by its original de f ining expression and using the fact that the limit of a fi nite sum is the sum of the limits, we have that (11) Placing the result (11) in the inequality (1) and rearranging the multiple su::-. we have Combining the inequalities (12) and (s) we obtain the inequality (3). 'Th is completes the induction. '1U'1-+ 0 with II ~1 II , and hence, that the right ~d side of (3) t ends to zero with jl .t:l II . Now since y , • • • ,Yn are Ci,Iltinuous, we have that 1 Therefore, by the definition of a multiple integral we have This completes the proof. ~be function u(r1 ,•··,tn) defined in equation (2) of the last · theorem we shall call the kernel of the multilinear form W. Using the formulas derived in the last theorem it is easy to prove: Theorem (23.22) The kernel u(r 1 ,···,t'n) of a multilinear form Wfy 1 , ·••,yrJ is of limited mu.ltilinear variation. jf Proot: We shall prove this result by induction on n. For n = 1 the theorem is a trivial consequence of (22.7), for in that case the · def inition of limited multilinear variation .coincides with the ordinary definition of limited variation for a line a.r interval, and by (22 .7), we V u = M. Assume now tha t for any multilinea r form Win (n - 1) (a}b 1 ) argunents Yi,·•·,Yn-1, the kernel u(t 1 ,···,tn-l) satisfies have where M is the modulus of the form W'. The functional v = v[y1,•••,Yn-l;?fJ is for ea ch value of ~n a multil i near form in Yi,•••,Yn· Hence for any subdivision An of the interval (an,~) and for arbitrary choice of the L = tl the expression W'5 ~J. L\nj v is a liliear combination n (an,bn) n n of v•s and is therefore also multilinear in y1 ,•··,yn-i· Moreover, using numbers fJ. • equation (4) (23.21) we have so that the modulus of w cannot be greater than M. given by replaci ng each argument y 1 by~ 1 ,m iter ated limit a s i n (23.21), equation (2). to be .L en Ll n u. (an, bn) Jn jn The kernel of w is (t;1f) and p4ssing to the . The kernel of i s thus shown Under t he induction hypothesis we have theref6re that Rearranging this sum we have, writing /J e ti• tP• ••!Jn, 7 118 Hence the induction is complete, and we have that the kernel u(-i-1 , ••. ,tf) of a form Wfy1 , •.•,yr} is of total multilinear var iation not greater than . M, the modulus of W. There is a certain property which .holds for functions of limited multilinear variation defined over an Hn, which functions, like u(z 1 ,··•,~) . used in the last two theorems, satisfy the condition of vanishing whenever one of the arguments, say )- takes on the value ai; namely, that if we keep certain of the arguments fixed and consider u(-z?-,••·,7?) q~ function of the others, it is of limited multilinear ~ these lat t er. We shall prove this in: Theorem (23.23) If u(z--1 , • • •, -z:-P) is defined for ~: (ai ~ ,ri ~ bi) and is of limited multilinea r variation VHnu over Hn, and if we have u = 0 when for any 1, T i= a 1 , then u is of limited multilinear var i ation over the hyperparallelopiped described b y any subset of the variables c 1 ,•••,c:-n, t he remaining v;:, riable being held . fixed; moreover the variation over this subregion is always les s than or equal to VH u. n Proof: We shall prove t h is by reverse induction on n. It is clearly of no moment which of the v~riables we choose to hold fixe d, since by renaming them we could put t hem in any preassigned order. r1 , then Let us first prove that if c:1 is given, fixed value u(i1,l,···,1fl) is of limited multilinear variation over the Hn- 1 of the variables r 2 , • • .,n. From the definition of VH IHn 1 LEjl • • .(:nJ. t>,. n we have for arbitrary . u n J1 .•• Jn I ~ VHnu a j 1 .j ... 6n· (1) If f 1 = a 1 we have u = O, and the theorem is trivial; if 1; 1 = b 1 , we take jl to consist simply of the two divisions a 1 and b 1 • 1 On summing out from . (1) the range (al, b) we get two tenns, one of which vanishes because of ..Jl u(a 1 , 0 2 ,···,~) = O, and the other of which is Taking For a 1 Ei = 1 we have from {l) <r1<b 1 we taketi 1 to have the point s of division a 1 , ,r, b 1 • Summing out the range (a1 ,b 1 ) from (1) we get (omitting the term which vanishes indenticahly) (3) On cnoosing e! = ± 1 and ci = ± 1 so that both the terms on the left hand side of (3) are non negati ve, we have (4) where e ~i = ± 1 are still at our disposal. The general theorem follows at once from this special case, by fix i ng one more of the va.riables of u ( -1 -r , 1:2 , • • ·, 'tn) and applying the argument we have ju.st given • . cJ, 6 It is true in general that if u( -r1, • • • ,-11) is of li:.Tiited multilinear va riation over Hn (a1 = i--1 = b 1 ) and if y 1 = y 1 (?!') (i = 1,2,··•,n) are continuous that the multiple integral exists and is a multilinear fonn in y ,·· ·,Yn• 1 We shall not prove it simply for the case where u is restricted, as in theorem (23.23), to vanish when ?f = a 1 for any i. The more general case will not concern us. As a preliminary, let us state the following lemma which is often used in the proof of t he existence oj an ordi nary Riemann-Stieltjes integral of a. continuous function with respect .,_ o a function of limited variation: Lemma (23.231) If f(x) is continuous on (a, b) and if /(x) is of limited variation on the same interval, and if ~ and Zi are t wo subdivisL. ns of (a, b) such that t:.. ~ F, then (i and ::\ denoting as usual any points in the 1 th interval of J and J' respectively. Th e proo f c omes out at once if we observe that A 1 1 is equal to L. Ajl extended over those subdivisions of K lying in the sun ( Xi,Xi+l) (xi,xi+l) of b. Thus for a typical te rrn in the first sum we h ave (2) which gives (3) Combining t he inequalities (3) for all i we have which is the inequality (1). A second leIDr.1a, which we shal l not prove explicitly is the following. Lerrrna ( 23 .232)Let F =L aj .. •j •~ •• • n 1 j ·••jn l a~ be a multilinear form with n i 1 . in the numerical components ej of n l·••Jn i . i 1 ordinary vectors e (i = 1,2,•·•,n). Then if I/IJ1fl denotes we J.k i . numeric a l coefficients aJ. ~ 1eJ.I, have Vi e shall now prove the following: Theorem {23.24) If u= u0"l"1 , ••• , -c; n) is d_efined over and h / J\i! (a1 fr,i ~b 1 ) of li:nited nro.ltilinear variat"ion over ~, if u = O when for any i = ai, and if Yi= yi(~ 1 ) (i = 1,2, ••• ,n) are continuous functions, ,22 1 2 n Le t t:. ■ ~ ·LJ ••• L1 and 6 = b-1 •L\T2 • • •!J-:on be two subdivisions of Hn arbitrary except that J ~ X- We wish first to establish the following inequality. n ;i M ~ IIY1ll •• •I/Y i-11• //y1+111 •• • J/yx:JI • ~1Y1 (-1) i] 1 represent points in the cells of ,1 and [i respsctively, and where .1 ~ z. where o~i and We shall use induction on n. For n = 1 the inequality (1) is precisely the inequality proved in Lemma (23.231). Let assume (1) for The subdivisions i::j and 3 and the numbers -r ~ and ~J; will be Ji i kept fixed throughout t he process; hence we shall denote y 1(~~ ) by y~ n - 1. Ji Ji and Yi(~ ) by y~ . Now v denotfng a function of~,··! ,rn and possibly Ji J1 i:f of other arguments we define ands}, by a similar expression, in which bars are written. The sums in the .left hand side of (1) are thus respectively s] u and S,a u. Let us first prove thay if we regard s~-lu qu! function of - l , then (2) 1 0 If c ~ = ± 1 we have Jn Talcing this last expression for j Fj in Lemma (23.232) and writing aJ. . l•••Jn = Aj 1•·· j nu, 8J~ i = y~ (i = 1,2, ••• ,n-1), el?- = ~~ we have Ji Jn Jn 1 L EnJ. ll~ csi- u) ~ M//y~/ ···IIYn-11 1 (an,b n Jn I (3) 0 ) Since this is true for every choice of ~nj . n s;;u = Now, observing that the equation (2) is established. I= ~j 4 nj ( s~- 1u), and using the (an,bn) n n Lemma (a, .231) with f(,n) E yn(~n), ,<1°) = sf- 1u we obtain (4) N0 w rearranging the sum in the first member of (4) we have L\ (a~bn) -n -n ( n-1 ) Yj 4j n n S4 u l n-1 =1 1 ,lfn-1 -:-n Yj • • •Yj Yj A Ji•• •4J·. _Aj u 1 ~ 1 n- n n-r n \ =L (5) Similarly rearranging sf u we have n-1 S;:-n u = S4 ~ £\ -n u~ L- -n yJ. hJ· ( a lljln) 11 n (6) i1f Subtracting (6) from (5) and taking the modulus (7) This last expression is in the same form a s the left hand side of equation (1) except that n is replaced by n-1 and u is replaced by 1 • • • ,1:n-1 less than MIY I n 11 • which we shall now show has a variation qua" r, Using Lermna ( 23 • 232) with e3 i • E' ] i ( i = 1, 2, ••• , ~l ) , e~n = Yjn' aj 1 •.• jn = Kj 1 ... Jnu we have (8) The induction hypothesis now gives from (7) and (8) that n-1 i1i M IIYnll 5_ II Y1il · •·I Yi-lll ·II Y1+1! •.. ~Yn-lj 41 Y1 (1..) (9) Combining the inequalities (4) and (9), there results (1) which was to be proved. From inequality (1) the existence of the integral follows at once. Let .ti 1 ,~ , • •• ,t\m, · • · be an infinite sequence of subclivisions of 2 ~ such that A 1 ~ Ll 2 ~ • • • ~ ~ ~ • • • and such that I/Amil+ 0 with ljm. //Yi/I being ·fixed and yi being continuous we have that ~ i y and hence 1 the right hand side of (1) tends to zero with ljm. Butthe inequality (1) is precisely the CO,Uchy condition for the existence of a unique limit for the sequence We sf1u, S~2u,•••,S1mu,···. Call this limit S. now prove that given e>O we may select d>O so that if jl.'111< ~ then _ [s~u - 8 / <.. e. _Suppose e given. Choose d so that if UDI~ ~ the right hand side of (1) is less than e/3. and Choose m so that 114J~ i !sfmu- sl<e/3. LetK:A+~.m Then sinceA~Zand4rt~$ _ we have from (1) and by the choice of m lsfm - s j<e/3 jsfu- sl< a or This completes the proof of the theorem and shows that Sis the integral. Theorem (23.25) The multiple integral defined in the previous theorem is a multilinear form in the continuous functions yi. Proof: bl bn In order to show that W= W(y1 ,··•,Yn) =)ai ( ...i'n( y1 (r1 )•••yn(r) du(t',•••,-z!1) is a multilineu, it is sufficient to show that it is linear and continuous in each of its arguments. The distributive property of the sum SJ u with respect to each of the functions y 1 is evident from the explicit form of as ll.611 ~ 0 of s:u Sfu. The limit is therefore also distributive. To show the continuity of a linear functional it is sufficient to show it at one point, say the origin. If we use LErmna {23.231), taking F = Snu, 9.i u we obtain Ji = Yji , a k j1·••Jn =~ j1···Jn allowing /Ill II to tend to zero we have which shows that Wis continuous at Yi= O. It 1s of interest to show that the integral defined in (23.25) can be calculated as an iterated integral and we shall find the result useful in proving unici ty of representa.t ion when the kernel u(-t1, • · • ,rn) is restricted to be regular. 1 h'\ Lll+t l1>\€CI r . Theorem (23.26) Let u(T ,··•,-(1) be of~limited variation on Hsi: (ai ~ r 1 ~ bi) and let u = 0 when t' = a 1 for any 1. Then for each r~n the iterated integral has a meaning and is equal to the multiple integral (2) Proof: Making full use of the notation introduced in .Theorem (23.24) we shall let S denote the multiple integral. The right hand side of equation (1) (23.24) is independent of the subdivision K. Letting 11s 1 ~o in that equation we have bl It is clear that the integral U br ejal ...) Iar y1 (-r1 ) • • •yr(-cr)du exists, for, by (23.23) u is of limited variation in ~l, ••• ,~r.. Let us rearrange the sum sf u (4) aDd place (4) in (3). Now b.Y the definition of a multiple integral, U is equal to the limit as 1141L12 • • •~ I +O li mit with L\ 1 , · · • ,t.r of sru. and leave Ll , ··•/in 141 If we pass to the untouched, the combination of (3) and (4) becomes (5) N0 w pass to the limit with the remainder of the~••• This completes the proof of the theorem. (5) then gives It may be observed incidentally that by a use of Lemma (23.232) we obtain The 1 imi t of ( 6) as!IAiAz • • •Ar !! ..,..0 shows that (7) Corollary (23.261) If u(-c- 1i···~n) is of limited I:QU.ltilin~ar variation over 1¾i, then U:: U(~) , Proof: ~_t ... ;;::\i··•Yn_1du is regular in -d1. Take U to be the U of the prec~ding theorem with r = n-1. Then· by equation (7) (23.26) U is of limited variation. The definition of U tells u.s that (1) and, writing out the formula analogous to equation (3) Theorem (23.26) in which we repla.ce S.6n u by Sn-1 u, S by U, we have 4 so that the approach is uniform with respect to -r;n, since the right hand side of (2) does not depend on -t1. Now since u is regular, we have for all~ lim h ➔ +O si-lu_(?fl+h) = n that s1-1u(tn) Since the limit in (1) is approached unifonnly with respect to ~nit follows that lim h++O u.(rn + h) = U(z;n) which was to be proved. Theorem (23.28) Two distinct regular kernels carmot represent the same multilinear form. Proof: We use induction on n. For n = 1, the result is well known, and is in fact a consequence of Theorem (22.67). For general n, let 1 Jl 1 J i'~+;~~t · u.("t' , • • •~ ) and uC--c- , • · · ,??) be two A regular kernels on iizi· We shall prove that tliere is at least one set of continuous functions Yi (i = 1,2, ••• ,n) such that f ~ Y • • • y du -/ 1 n ir Yi · · • y n du (1) By Theorem (23.27) and (23.271) the integrals in (1) may be written respectively as where U and dU are defined as in (83.27) for the regular kernels u and u respect.ively. then U = U. Now since U and U are regular it follows that if for all Yn Now using the induction hypothesis for n-1, it follows that since U and U are identical forms in Yi,•·•,Yn-l' we have u = u, which contradicts the original hypothesis. .!. 130 24. ! Theorem 2.!!, the Limits of StieltJes Integrals In this division we s hal l give a proof by elementary methods of •a generalization to Riemann-Stieltjes i ~tegrals of the well known Arzela theorem on sequences of Riemann integrable functions. Let us first prove several lemmas: Lemma (24.l) I f , is bounded and monotone increasing on (a,b), f is an · function on (a, b), and if A. and~ are subdivisions such that 4 c. jj, then and . in general if J CL\, (2) whereW1f is the oscillation off in the interval (x 1 ,xi+l Proof: Since we have (3) Summing (3) over all intervals in J we obtain (2). If J = (a,b) we have (1}. 13 1 Lemma (24.2)· If~ is bounded monotone increasing and if f is any b function on (s,b), then in order f d~ exist it is neces sary and J a sufficient tha t L w if A1 ~ ➔ O with~A{~ (a, b) SufficiencY: Sel -ct a sequence of subdivisions that /\ Lt [( ➔ 0 and such that tf C. l A. nf such 2!"1 • Denote b y SA and S.4 th ~ expres s ions occuring in the left hand side of inequalit y (1) of the preceding lemma. Thus and by hypothesis, the right hand side tends to zero with 1/n. · s eque~ce The t SAn ) therefore has a 1 i mi t. Calli t S. If e >0 ~ cho ose S so that if fl A\\ <. o ·L u,; 1 f ~i~ < ep,.· Choose n so that \ s6 n 0 Then if A is any subdivision sat isfying Zl CE +An 0 - S \ e/3. I\A II~ b we h.aae, since and 6n C.~+ll \ S ti - i +An \ <e /3 I Sc,n - 8t, +An j <e/3 \ s-s}< e So t h at Necessity: For a ny division let fi and fi denote respe ctivel y the upper and lower bounds of f(s1) in (xi,xi+l). 6 so that \\AI l <. 6 i mplies then Given e) O, select j Lemma (24.3) Let f,g be integrable with respect to bounded mone- t .one i ncreasing !- Let m(f,g) de~tre the function whose value at ea ch xis the greater of f(x) and g(x). Then rn(f,g) is integrable with respect t o ~Proof: Since ff di and Jg d¢ exist we have ~hat ·,,. with \ All . Hence by the 9r eceding lemma ) m( f, g)dl exists. Lemm& (24.4) Let f1 ? f2 ~ · • • = fn => • • • => 0 be a sequence of function s having Oas their limit. lim ( fndi = 0. n-+0 ) Sel ect e ) O. Then if [fnd? exists for all n, ·Let ~ en l be a sequence of positive terms such that L, en converges and is le ~:; s than e/4. Select t:.1 (. ti ( · · .tf c_ . .. a sequence of divisions sucn that (1) ~ow with each point x of (a, b ) we may associate a closed interval containing x, as follows: Given x, choose nx so that fn(x)~ef2V(a,b)I• Let Ix be the interval (or intervals in case xis an end po int) of 6nx which contains x. '3 2 We now apply the Heine-Borel result and establish the existence of a finite number of points x.., • • • ,xk, such that I l Xl •••I- entirely Xk cover (a,b). i.rrang e t h e distinct integers of the set (n~ , • • • ,nXk) in order 1 of increasing mangitude, say n1 ,··•,n 8 • Let J .fow if 1 denote the sum of all intervals Ix·1 for which n..l = n-. Xi (x_,,xj+1 ) is one of the intervals of Anl lying in J , it must, 1 . in order to have b r:•en i ncluded in some I, contain at l 3ast one of the . points x1 , • • • ,Xk:· Write (1) with the points §1 lying in J 1 replaced by memb ers of the set xi, • • •, xk, and the others unchanged , and add the resulting inequalit y to ( 1). There results which now ~ust hold if in particular f(J 1 ) are feplaced by f • 1 Hence we obtain n ow take for J 2 the sum of the s e t of I's, t r1ose I- Xi but wa ich are not included in J 1 . for which Dx· = n , has, as before, one point of the set ij_,···,~· 2 l Each interval of t{12 lying in J 2 By a similar process to that used above we obtain the inequality continue t ::lis p rocess s times. Jl' J 2 , • • • ,J 8 completely cover (a,b) - and are non-overlapping except for end points . iuiding the results, we have Ti1is proves the lerrrna. Corol l ary: If the sequence f ,f2 ,•••,fn,•·• is monotone 1 increasing and converges to zero, a similar theorem holds. We are n · w in a position t o prove the theorem: Theorem (24.5) Let~ be bounded monotone increasing and let lfn~ be a bounded sequence converging to zero. Let ) fnd~ exist. Then lim ) fnd~ = 0 n ➔ oo Proof: Define , for k 1 n, funct i ons Ynk !! Ynk(x), Ynk = ynk(x) S0 that Ynk is an increasing sequence ink and ynk is decreasine ink. The _se quence of numbers ar e therefore respectively increasing and decreasing and are bounded. Let Bn be the upper bound of fb Ynk!Ji and bn t_he lower -bound of / 'ynkd~. a We have, using the fact that 9 is monotone increasing, a Mow since Yn+p,k = Ynk we have :Sn+p ~ -:Sn and similarly bn+p ~ bn . If therefore we can show that: (I) :Sn~ 0 ~ bn (II) lim (Bn - bn) ~ 0 n ➔ oo the theor em will be proven. To prove (I), define Znk(x) = les ser of Ynk and 0 znk(x) • greater of Yni<and 0 f bn ~ f lln ~ so that and ; y l)kd~ ~ f¼z:_d~ ynkd~ ~ h1N But~ is monotone increasing ink and has O for its limit. By the corol l ary to Lemma (24.4) we have and similarly by ( 24 .4) itself' lim k ➔ oo r Znk_-0.1 = 0. This proves (I). To prove (II), let e be given. Select a seqvence ~erJ of positive nu~bers such thatL.en converges and is less than e/4. and Choose rn so that • for all k ~ rn. N0 w define: l east of (Y1r , Y 1 < 1¾i+e· Then certainly Hn > = Hn+e and I1n = 2 r2 ,·••,Ynrn ) Let k :>r1 ,r 2 , · • • ,rn• From the definitions we have that (i = 1,2,•••,n) and that Ynk - ~ is less than or at most equal to at least one of them. Therefore Integrating with respect to cf and alllowing k to increase without limit we h .- ve By a similar argument • But Hn - ~ is monotone decreasing and its l imit is nowhere positive. Hence defining Ku= greater of (Hn - 1¾i) and 0 we have that and applying Lerrana (24.3) the ri ght hand. side tends to ef2 as n~oo. Thus lim n-=tm (1\i - ¾) = O. CONCLUSION AND SUMMARY In part 1, vector spaces with closure under multiplication by real and complex numbers were defined. Polynomials were then defined and discussed for both types of spaces, and it was shown that for spaces E(R) the definition given was equivalent to that of Frechet. A special result of some interest was given on the continuity of a multilinear form in the ensemble of its variables. Modular properties of polynomials were discussed. In division {14) application of these results was made in the definition of analytic functions in vector spaces. division 1a essentially virgin. The field of this It was pointed out that a distinction arises between spaces E(R) and E{C) in the methods necessary to prove certain theorems on analytic functions. Various theorems of a fundamental nature were proven for analytic functions on spaces E(C) and a fewer number for those on spaces E(R). It was shown that the generalization of the completely integrable Pfaffian System has, in spaces E(C), a unique analytic solution. In part 2 there was given a generalization of Frechet'a theorem on the representability by means of double Stieltjes integrals of the general bilinear continuous functional form on the space of continuous functions. It was proven that a multilinear form inn variables is repre- sentable by means of a multiple Stieltjes integral and that the kernel ot the integral if regular is unique. A proof by elementary methods was also given of the generalization to Riemann Stieltjes integrals of Arzela's necessary and sufficient condition for the term by term integrability of a sequence of Riemann integrable functions. In conclusion, I wish to express my thanks to Professor A. D. Micha.l for his suggestion and guidance in the construction of this thesis.