Integrals and Functional Equations in Linear Topological Spaces Citation Hyers, Donald Holmes (1937) Integrals and Functional Equations in Linear Topological Spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/dstq-7n64. https://resolver.caltech.edu/CaltechTHESIS:11252024-232906027 Abstract No abstract. 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 1937 Record Number: CaltechTHESIS:11252024-232906027 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11252024-232906027 DOI: 10.7907/dstq-7n64 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16880 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 26 Nov 2024 18:17 Last Modified: 26 Nov 2024 18:34 Thesis Files PDF - Final Version See Usage Policy. 25MB Repository Staff Only: item control page

Integrals and Functional Equations in Linear Topological Spaces Thesis by Donald H. Hyers In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1937 Resum~ The classification of linear topological spaces is discussed. In particular, it is shown that locally bounded linear topological spaces are pseudo-normable and that locally compact linear topological spaces are finite dimensional. Spaces of type (F) are characterized among the class of linear topological spaces. A generalization of the Lebesgue-Frechet-Birkhoff integral is carried out, and Riemann integrals are discussed. By introducing suitable systems of sets, called K-systems, it is found possible to extend the method of successive approximations to functional equations in linear topological spaces. for K-systema is proved. An existence theorem Existence theorems are then obtained for the functional equation y = f(y) and for first order differential equations in linear topological spaces. A ·· b egi-nning-rs--ma-de-en-&=g&~-&'.t~~~'BZ"'ffle~...001' -%mp;ke,.teJ:~ e'<:mirhlue.tts~r: : f l ~ ~ e n ~ ~ s . Table of Contents Chapter 1 Linear Topological Spaces Page Sec. 1. The calculus of sets in a linear space 1 2. Linear topological spaces 4 3. The postulate of convexity; relations between the spaces :J and Neumann's spaces L ' ]0 4. Sequential convergence, the postulate of completene's s 13 5. Topological boundedness and compactness 14 6. Continuous isomorphism; relations between linear topological 7. spaces and metric spaces 18 Realizations of linear topological spaces 29 Chapter 2 Integrals in a Linear Topological Space 1. Introduction 3fl 2. The domain of the independent variable 38 3. Unconditional convergence of series in T 40 4. Lebe sgue integrals 46 5. Riemann integrals and continuous functions 55 Chapter 3 Existence 'Iheorems for Functional Equations in Linear Topological Spaces Sec. 1. Page K-systerns 63 2. 'Ille equation y = f(y) 69 3. An existence theorem for a differential equation 70 4. An application of the general theorems 78 5. Completely continuous · linear transformations 80 Appendix I: 'Ille Kuratowski Postulates for a Topolo.gical • Space 87 Appendix II: Pseudo-normed Spaces 89 References 91 i Introduction A linear topological space may be characterized as being a linear space supporting a Hausdorff topology in which the operations of vector addition and scalar multiplicati on are continuous. A. Kolmugoroff, J. v. Neumann and A. Tychonoff were among the first* to study such spaces. Kolmogoroff in 1934, introduced the useful idea of boundedness in linear topological spaces, and was able to characterize the normed spaces among the class of linear topological spaces as those having the properties of local convexity and local boundedness. Neumann in 1930 had used several instances of linear topological spaces to good advantage in the study of functional operations in Eilbert space (cf. Neumann ( 1.)). In 1935, Neumann studied the notion of completeness in a linear topological space L, and in collaboration withs. Bochner, investigated almost periodic functions on an arbitrary group to L (cf. Bochner and Neumann (1.)). Tychonoff generalized the Brouwer-Schauder fixed point theorem to locally convex linear topological spaces, and gave a new way of defining the "topological product" of a collection of topological spaces. A. D. Michal and E.W . Paxson** were the first to define a "Frechet" differential for functions with arguments and values in a linear topological space. They also developed a theory of Riemann inte- * Kolmogoroff (1.), Tychonoff (1.), Neumann (2.), where the numbers refer to the list of references at the end. ** Cf. Michal and Paxson ( 1 . ) and ( 2. ) ii grals for such a space. The first chapter of this thesis, after the first four introductory sections, is concerned mainly with the classification of linear topological spaces (Begun by Kolmogoroff). The property of local bound- edness is studied, and locally bounded spaces are found to be "pseudonormable" on the one hand, and metrizable on the other. Frechet (F) spaces are completely characterized among the class of linear topological spaces. The main resiilt is that every locally compact linear topological space is finite dimensional. In the last section of the chapter a number of examples of linear topological spaces (all non-normable) are given, illustrating various theoretical points. Chapter t wo is devoted to a generalization to linear topological spaces of G. Birkhoff's integral for functions with values in a Banach space, and to a brief discussion of Riemann integrals . In the last chapter, on functional equations, certain systems of bounded sets, called K -systems, are introduced, making possible the extension of the method of succes~ive approximations to functional equations in linear topological spaces. A method of constructing K -systems for~ locally convex linear topological space is given. Existence theorems are proved for functional equations of the form y = f(y) and for first order differential equations. In· -the-- last--·secti.-on~of"iihe chapter completely..--eont4-ru:x€mB---±-:kle~...ans4!e'F-Nl,8:t;i.,o~4-Roo-.,,.£er ~ ~ :.,;t,.Pporogicctl--:-spa:-ce."B·; anfr---a ..•beg..ia11si.ng--i.s.,;.made,...on....a-g.e.-~za.t-i-en-G-f 0 ..;;Bies-Z-·- theory· or- comple-te-:1:y c-cmti--nuo11~---li·n&ar- -tra:ns-f-ormati-ena--. iii I wish to take this opportunity to express my thanks to Professor A. D. Michal for bis inspiration and guidance. 1 1 , ;¥1 Chapter 1 Li near Topologicai Spaces The cal culus of set s i n~ linear space 1. Defini t i on 1 . 1 A syst em consi s t i ng of an abstract set the set "+j and OCJt R of real numbers °' , ,-. ~ --- I of el ements and operations i s called a linear soac e i f t he f oll owi n g po stul at esl(- are sati sfi ed: ( i ) 1l + 1-" f , . E T (ii) Ot JC = 'f + X (ii i) J< -t- '( (iv) ,I( -t- ( 1 +..z) = (~ +- 'f-) + .z. . :> . (v) x+ 1-- . =x+z ( vi) cc (JC -t- f) = o< I< -f- 'f (vii)(« -t ~ } X = « x + f-i x ( viii) ~ ( (l;c) = (o< (9 ) X (ix ) i X X - 1- =z Of It i s well knovm that these postulates i mpl y (1 ) t he exi s t enc e of a unit 6 and an inverse - ;< = (-i) X i mplies « = ~ °'- :f: 0 . for x I= 8 for the operation + , ( 2) tha t o<Jt = f3X and ( 5) that "' " = ot "f i mpli es x =~ f or A linear s pace T wi l l be call ed finite dimensional i f ev ery el ement is uni quel y e:;-q)ressi bl e i n the form oc , ~ ... • • • -t-atl'XI" f or a f i xed f i nite set ( >C,, • • · , X,...) c: 'T' , ru1.d inf inite di mensi onal i f it i s not f inite dimensional. ➔~ The theory dev eloped i n t hi s thesis appli es equally well t o Cf. Banach ( l.) p . 26 . The si gn = denotes logical i dentity. For an independent set of po stul ates for a linear sp a ce in whi ch = is an undefined rel ation see Taylor and Highberg, Comptes Rendus of t he Warsaw Society of Sciences, vol . 28 (1935) , pp. 136-142 . 1, #1 2 finite or infinite dimensional linear spaces. By a function £!!_ a set P to a set Q we mean a rule which assigns P one or more elements of Q . to each element of functions, where to each ;< f P For single valued there is ordered just one element of Q , we use a small letter such as {(x) for this element. In the case of functions which are not necessarily single valued, the set of values cor- • responding to X will be denoted by a capital letter, viz. F(x) . In all cases the set P will be called the domain, and the set of correspondents of the elements of P the range of the function. set of the domain P of a function F numbers. , the set of values M is any sub- f F(x)}, IC ( M F{M} . will be denoted by Let If 5 , S, , s:& be subsets of 7' , and let A be a set of real In accordance with the last paragraph we denote by J< + S the 'er where -<f ( 5 (read j is an eleroont of 5 ), by S, -t- S., the set of all elements t- i- where ){ E 5, and 't € 5 ..,. , by set of all elements ;c + JC °' 5 the set of all ot X with .tJtf f S and by A5 the set of all with °'~ A and x f S . Finally we write Cr5 .,, c:,c x for the "convex hull" .5 , that is the set of all finite sums =~ at where ex~> o ~ '::.' ~ I ° ' . , . = 1_ , Xe, (; 5 , and -P runs over all the positive integers. of ;< .,, c. )(,. '-"'• A subset 5 of T will be called convex if ~ S + (1.-«)5 =S Definition 1.2. for o < °' <- 1 . We shall make considerable use of the calculus of sets in linear spaces . For convenience we quote the following results in this calculus from a paper* by G. Birkhoff. * Birkhoff (1.) pp. 358-360 1, #1 ( a) 3 The following pr,operties of linear spaces hold for the vector su.ms of sets: V 1 SI+ - ( S 1 + 5~) + 5 3 s, -t- S";J.) - o(. o( ( V4 0(. ( ~ ( c) ( d) = S,_ -t' s. V 2 5. + ( 5 1 + S?J) V3 (b) S,._ S) S, + °'- s .. - (« f3) s - s 5 V 5 .1. V 6 s+ e - s S is convex if and only if • ~ S = S . The operation ~ S is additive and homogeneous that is c,. (s. + S:a.) = ~s. +Gr~ and Ce-(« S) = ex: e,5. C,. S is the smallest convex set containing 5 . Besides the above vector operations on sets we shall of course make use of the usual logical operations, and these will be distinguished from the vector operations by the use of a dot. denote the union of S, and S a. , 5 , 1 S, .;_ S~ the set of x '.s Y. not in 52. ) • 5 1 • S2. for which Thus 5, -i- 5:a. the intersection of J< E S, will S 1 and and x. E S.:i- (read The empty or null set will be written e . It will be convenient to denote the linear space and the set of all elements of the space by the sa.11l.8 letter T . properties of convex sets. If .5, sets «. S, + (3 S 1 and For future reference we add ·the following and S. are convex, then so are the S 1 • 5,.. , assuming that 5, • S~ .::f:. 0 . This statement is obvious from properties (b) and (c) and definition 1.2. Again, if -f(x) with domain '"r. and range .:_ is distributive, i.e., if 1, #2 4 f (C(' 1' + '31--) = °' t (;c) + ~ f(.f) then the set -f-(S) o·f for all ><., t E 'f, and all real °", f3 '1'1 is obviously convex whenever S is convex. We may sum up these statements by saying that the property of convei ity is invariant under the operations of "linear combination", "logical multiplication" and "distributive transformation". A linear space L which is a sub-space of a linear space ~ (with the same operations defined in both) will be called a linear manifold in 'T' . The smallest linear manifold L containing a given set S will be called the linear hull of by .5 . 5 , or the linear manifold determined The linear hull of a set S consists of the set of all elements where Jt, f S.., a" is real, and ~ runs over the positive integer .S. 2. Linear topological spaces. Definition* J .3. A system J consisting of a linear s:i:ace T and a family of point sets U of T will be called a linear topological space if the I following postulates are satisfied: (I a) For each point JC (I b) For every pair Ux ' ( I c) If -f € UJC ( I d) If ;< :f: (II a) Given any Ux +- t of 'l' there is a u: Ux with x € Ux . there is a u;' such that u; C U,x. u:. there is a Jf there is a U,x not containing -f . there exist U.x and U1 such that U.x + U c Ll.x.,.. . 1 0 * This definition of a linear topological space Js equivalent to that given in Kolmogoroff ( 1.) , in which the closure S of a set .S is taken as an undefined notion. See Appendix I. 1,#2 (II b) 5 Given any llx there is a Ux mid-point"' such that I« Ux c ~nd a real open interval I .. with Uocx The sets Ux are called fundamental~, or neighborhoods, and postulates (I) furnish a neighborhood topology for our space. (I a), (I b) and (I c) are too three neighborhood axioms of Hav,ddorff and (Id) is the first or Frechet separation axiom. The linear space T basis of the linear topological space cf . will be called the A given linear space T may be the basis for various linear topological spaces~ J," · ·· corres- ponding to different choices of the family of fundamantal sets. The following definition is a statement for linear topological spaces of the Hauaidorff equivalence criterion*. Definition 1.4. Two linear topological spaces J basis 'r and with families f Ux J and f V.,. J and J' with the same respectively of fundamental sets will be called equivalent if for each point x and each Ux there is a VK C of 'T1 Ui, and for each ~ there is a Ux c Vx. On the basis of postulate (I a) we make the following definitions. A point x is· called a contact .point (beruhu.ngspunkt) of a subset S I if every- Ux contains a point 1' f 5 . 't :F X of If every Ux contains a point 1 c5 of S . The closure 5 of 5 is defined as the set of all the ¢.oMta.ct with then x is called** a limit point (haufungspunkt) * See Alexandroff and Hopf (1.) p. 31 ** In spaces satisfying (Ia) and (Id) it is clear that this definition of a <kic1t1·i t ":i on point is equivalent to the following " X is a li.-fft it point of S if every Ux contains an infinite number of points of S . 11 1,#2 6 points of S , and a set is said to be closed if S c 5 . An open set is defined as the complement of a closed set. '.Ihese definitions apply not only to linear topological spaces, but to any space linear or not, in which there is a neighborhood system { U ing merely ( I a). E , ! satisfy- The following theorem gives us an equivalent defini- tion of an open set in 8.DY such* "umgebungsraum" E . If £ is any space with ~ neighborhood topology satisfying Theorem 1.1. ( I a) , then ~ non-empty set 5 of E ~ open if and only if x € 5 implies ~ there exists ~ Proof: Let UJC c 5. 3 be open and let x E S £. ..:. S every Ux would contain an element of the complement E .:-5 would not be closed. 5 Then no JC € such x. there is a neighborhood Corollary: Conversely, let JC e 5 can be a limit point of u}C Ux c 5 Then if there were no £. ~ 5 imply , and Ux C5 . , since for any containing no point of E -=- 5. If £ has ~ neighborhood topology satisfying both ( I a) and (I c), then every neighborhood UJt _!.!~open set. In particular every fundamental set of~ linear topological space is open. From this corollary and the definition of open sets it follows at once that the three following characteristic properties of open sets hold for any umgebungsraum satisfying (I a), (I b) and (I c), and hence are valid for any linear topological space. * (i) The null set and the set of all points of the space are open. (ii) The union of any number of open sets is open. (iii) The intersection of two open sets is open. These spaces are discussed in Alexandroff and Hopf (1.) p. 30. 1,#2 7 ' We now extend the concept of "neighborhood" by the following definition (cf. Hausdorff (1. ) p. 228). Definition 1.5. A family of sets V will be called a complete neighbor- h22d system for the r n if (1.) each member of the family is open (2) for any point X and any fundamental set U, there is a member of the family with X f V c:: U,.. . A family of sets V V will be called a complete neighborhood system of ~ ~ Xo if (1.)' every V is open and contains Xo (2.)' for every U,c there is a 0 V c:: Ux 0 . Using the above corollary it is easy to show that any complete neighborhood system satisfies postulates (I a) - (Id), and that if any such system be taken as tha family of fundamental sets of postulates (I) the resulting linear topological space is equivalent to the original s~ce in accordance with definition 1.4. The family of all open sets of the space constitutes a complete neighborhood system for the space which is in f'actthe largest such system. Now consider some consequences of postulates (II). If f (}() is a function with ugurnents in a linear topological space 3 and values in any other linear topological space ;/ ' , we say that f (JC) at the point X if for every u ;(JtJ there is a U K is continuous * such that f( U.ir ) c::: u ;(1'). A similar definition applies to functions of more than one variable. We now see that the postulates (IIa), (IIb) state merely that the functions X + 1 and « X are continuous. Since· in a linear space -)( = (-.1)X it follows that the "inverse" - JC of x is a continuous function of x . * For other equivalent definitions of continuous functions in topological spaces see Alexandroff and Hopf (1.) pp. 52-53. 1, #2 8 Using postulates (II) and the criterion for openess given by theDrem 1.1 we shall obtain Let G , G-, , G-2, be non-empty open sets of _! linear Theorem 1.2. topological space cJ , A ~ non-empty open set of ~ numbers, real number ,;: 0 , and -'f ~ point of J ~ all OE A open sets, e E G- or Moreover if either of the conditions holds the set AG Sinee G is open, Proof: X f just proved, with ~ = -1 ,t E G-,_ u"t C u~ ...1- - V,, C U,r C G; open. LJ~ -t-1" Now take is open. But - Similarly -'f ± G- there is a real interval I« UJC C . Hence ") ~ Ux c and LJt such that and a v.,. C u1' • G-,._ . Then Ux +i C: ~Ux + u"t C G-; + G._ G;. is open, so G. - G-2.. is open. of .A UJC cG-. By postulate ( II a) and the fact and hence theorem, first suppose a. C u~ C: G , it follows that given Ux+t - so that G; -t- G-.._ implies t he existence of a is open. there exists a u}C • G is open. ~ with Vt1 v... By postulate ( II b) there is a and (3 G , ~ ± G-,_ Then (3 G- . f' ~ . is also To prove the last statement of the Then for any Ol f A with center « such that G- . By hypothesis ex /: 0 and. any I., c. A x E G, and , so there is a r eal interval I~ with ½~ c l.« c I\. By postulate ( II b) there is a real interval u(J{X such that Iq u a neighborhood r ,,V« and , ,, C' I I V,.r • If On the other hand, without requiring OE A ~ IvcaG c:: I·~ U« }C C • Iv., we have 1/I,{.; Ux whence AG is open. o< x C I« U;r C AG-, 1,//2 suppose that (::) E G- • Let ot <= A the above proof applies and yields •U,u c. A G. show that there exists a Since 6 E (; Ue c AG-. , there is a Ve c G-. is a Ue with Y(J Uec. Ve c Gor Take If « = O , we must (3 E A , 13+0 But by postulate (IIb), there U8 c:AG. From this theorem and postulate (II a) it follows that the family f JC + VJ where X runs over the points of J and V runs over any complete neighborhood system of the origin (or any other point) is a complete neighborhood system for the space ;J . This fact, familiar from the theory of topologi cal groups, is very useful in arguments involving limits. The postulates (I) taken by themselves are weaker than the postulates for a Hausdorff space because (Id) is weaker than the second or Hausdorff separation axiom. However, postulates (I) and (II) taken together do imply not only that d i s a Hausdorff space but indeed a regular Hausdorff space, that is, the second and third separation axioms* are both satisfied. The following theorem stating this result is proved in Kollnogoroff (1,), and is shown to hold also for topological groups. Theorem 1,3. A linear topological space 3 _!.! ~ regular Hausdorff space. That~ the following conditions are satisfied: (1.) )( ~ j u~ • u--r = (2.) 0 For~ Ux there is~ neighborhood is contained in Ux, U't with implies that there exist , U.x whose closure Ux . * cf. for example Alexandroff and Hopf (1.) p. 6?. 1,#2 10 Thus we may characterize a linear topological space as being (a) a regular Hausdorff space (b) a linear space in which the operations of vector addition and scalar multiplication are continuous. The linear topological spaces which have been the most studied are the linear normed spaces, where by a linear nonned space we mean a linear space T in which there is defined a function IIXII on 'r to the real numbers satisfying ( i) II JC II ~ 0 (ii) Ile{ x ll ( iii l II x .... 11 x II = 0 implies K - 8. = lee.I II x II . 1 n ~ 11 x n ..- u 1- n . It is easy to see that the postulates of definition 1.3 will be satisfied 'I by any linear normed space if we take the I( 1 - ,. If < '/,, , ,,=,~2.~ --- . LJJC S to be the spheres The relation between linear topolo- gical spaces and linear normed spaces will be discussed in section 5. In section 6 a number of examples are given of linear topological spaces which are definitely not normed spaces. 3. 'Iha postulate of convexity; the relation between the spaces d and Neumann's spaces Definition* 1.6. L . A linear topological space cl will be said to be locally convex if the following postulate i s ...cill:f·••·d : ( III) If U is ~ neighborhood of the origin 6 there is ~ set Y ~ taining 6 and contained in (1.) JC f V U such that implies the existence of a ult with u II C: V * Locally convex linear topological spaces are defined in Tychonoff (1.) p. 768. 1, #3 11 (2.) implies that 0<aG<1 °' II + C.t - GC) V= V (that is V is convex). An equivalent statement of (III) is that there exists a complete neighborhood system f V} of the origin with each V convex. It is this postulate of convexity which often allows us to carry over useful methods of proof from linear normed spaces to linear topological spaces. This is usually because the laws oc. and V + ~ V =c« +f.JJ V °' V c f5 V hold for convex sets taining o < ~ < f3 for V containing 0 , but not for arbitrary sets con- e. We next consider Neumann's definition of a linear topological space in relation to ours. In Neumann (2) p. 4 a linear topological space is defined as any linear space L for which there is a family U of sets U satisfying the following postulates Uf U {l.) If then (2.) There is a sequence 8 E LJ . u,~U,.~--- of sets of -U whose intersection is the single point B . (3.) If ~ Ve U (4.) If U E 1l there is a implies (5 . ) "' -V c U· If U EU there is a If We U with We U· V. V € U such that -1 ~ oc. ~ 1. there exists V f: U with · V -t- V c U ff L Uf: cl there is an a: with -f € «: U . ~ If in addition the postulate 1, #3 12 ( 7. ) If Ue U , then U -t- U c:: ~ U holds, he calls L a "convex" linear topologi cal space. Neumann has shown that a space satisfying (1.) - (6.) is a regular Hausdorff space in which the operations of addition and scalar multiplication are continuous, so that any linear SI)ace L satisfying (1.) - (6.) also satisfies our postulates (I) and (II) Ui denotes the ( choose U ¥ -= X +- U;. •where interior of U) so that our definition of a linear topological space includes that of Neumann's. We shall now show that Neumann's "convex" linear topological spaces are likewise included in the class of what we have called locally convex linear topological spaces. Theorem 1.4. Any linear space satisfying (1.) - (7.) ~~locally convex linear topological space. Proof: Let L be a linear space and -U a system of sets of L satisfying (1.) - (7.). Then, as we have already remarked, L if we set U¥ =X + l} i. . We now show that (III) is satisfied. Neumann's theorem 12, loc. cit. p. 9, states that U E 11,, satisfies (I) and (II) implies that the closed hull LJ, of U, is convex. Again,given any U e U a such that W,· c fh., since L is a regular Hausdorff space by W f U theorem 1.3. Hence for any. U fU - there exists a W there exists ~ U such that = W·' c:: U;_ . Denote Gr W,; by V . V is convex so that condition (2.) To verify that V also satisfies con- _O bviously of postulate (III) is satisfied. dition (1.) of the postulate, let X x = .z '"'• ~c. Xe. be any point of , where a:::."' > 0 ~ V. ...Z:a. =.1 , '"' 1, #4 13 , , U € U so that X, + U;, ~ ~ and then choose W such that , , Gr W,; C Ul . On using the obvious equality 1 Choose we obtain Cr W,; CU· <, • Thus condition (1.) of postulate III may be satisfied by taking U JI = X +- W-' ' , so that any linear space for which (1.) - ('7.) hold also satisfies (I) - (III). 4. Sequential convergence; the postulate of completeness. Following the usual definition for metric spaces we shall call a sequence of points of ;/ convergent if there is a point " such that corresponding to every nei ghborhood for all writing lim .,,_.. X-,, =X . 1, ')' V of 3 LJ~ there is an integer -,, This statement is abbreviated by It follows at once from the Hausdorff separa- tion axiom (see theorem 1.3, (1.)) that a convergent sequence has a unique limit X. In metric spaces it is well known that any limit point of a set is the limit of at least one convergent sequence out of the set. This property however does not carry over to an arbitrary linear topological space, when the first countability axiom* is not satisfied. In fact Neumann ((1.) p. 380) has given an example of a linear topological space, namely Hilbert space with its weak topology, in which there is a denum- * Hausdorff (1.) p. 229, axmom (9). l/14 14 erable set of points having the origin as a limit point but containing no sub-sequence which converges to the origin. This fundamental difference between topological spaces and metric spaces has recently led several authors* to the consideration of various definitions of completeness for linear topological spaces. We shall have occasion to use only the notion of sequential completeness (see Neumann, loc. cit., p. 10) which is a direct generalization of the usual notion for linear normed spaces. It appears to be a weaker condi- tion on a linear topological space to require sequential completeness than to demand completeness in the sense or Neumann's or Birkhoff's definitions. Definition 1.7. A linear topological space ; / will be said to be sequen- tially complete if the following postulate is satisfied: (IV.) ,!&1 X, , x,..) • - - be ~ sequence of points of ;;f with the property U of the origin there is ~ positive integer I'- ... f" (U) such that 'I,). > f" implies x$ - x,. E U that corresponding to each neighborhood ~ there exists ~ point ~ there is~ integer 'Jl=1'(~) ~ of ;/ such that corresponding to every that L ,..,, implies Ux Xe. E Ux. We shall call a sequence fundamental if it satisfies the condition stated in the first sentence of postulate (IV). This postulate may now be stated in the familiar form: ttEvery fundamental sequence is convergent.tt 5. Topological boundedness and compactness. The idea of boundedness of sets would seem to be entirely foreign * Neumann (2), Birkhoff (3), Graves (1). 1,#5 15 to topological spaces in which there is no metric. However in the case of linear topological spaces the operation of scalar multiplication makes it possible to generalize the notion of boundedness from linear normed spaces in a very useful way. Three superficially different definitions* of boundedness, all using scalar multiplication, have been given recently by A. Kolmogoroff, J". v. Neumann, and A. D. Michal and E.W. Paxson. Definition 1.8 a (Kolmogoroff) will be said to be bounded if implies lim Q',• X c. = 8 .. ➔ 00 Definition 1.8 b (Neumann) A set Sofa linear topological space °'• real, lim ~~ = O , X, E S A set Sofa linear topological space will be said to be bounded if for any neighborhood U of the origin there is a number °' = <=-:: ( U) such that °' S C U ., S of a linear topological space will be said to be bounded if for any ¥ ES there is a positive number G=6 (/ro) such that Xo is not in S for any« greater than 8 Definition 1.8 c (Michal and Paxson) A set 0 0< in absolute value. Michal and Paxson have shown that their definition of boundedness is equivalent to Neumann's. If we can demonstrate that Kolmogoroff's defin- ition is also equivalent to Neumann's, it will follow that all three definitions fall together. Theorem 1.5. * Definitions 1.sa, 1.8b, and 1.8c define t h e ~ class of Kolmogoroff (1.), Neu.mann(2.), Michal and Paxson (1.). The definition 1.8 a was given for t he case of spaces with a metric topology by Mazor and Orlicz, Studia Math. IV (1933) p. 162. 1,#5 16 Proof: Let* Sc T be bounded according to definition 1.8 a, and assume that S is not bounded according to 1.8 b. there is an X ES with Of .K e T -=- U of numbers which approach O with 1/f;. with Xe. and ~s s uch that a:::,.)(,_ e T 3 is not bounded by 1.8 a. Then for some U and each ex/. o be a sequence ..:. u ' so that lim. ,~- «c. )(,_ Conversely let i= a > S be bounded by 1.8 b. Let U be any neighborhood of the origin and. let f«-., sequence converging to O . f J< c. } 'l'hen there is a sequence By the continuity of J C(' ;t be any real we can choose a 8 >O and a neighborhood V of the origin such that '°' / <. 5 i mplies «: V c:: U . But by 1.8 b there is a real number f3 so small that J<, E 5 implies that (3 Xi. € V Choose -v so large that f /-< ~ ;c. for <- > -P so that 'l'hen t. '> Y implies that S is bounded by definition 1.8 a. theorem 1.5. 'fhis completes the proof of Besides the equivalent definitions 1.8 a - 1.8 c it will sometimes be convenient to use the following which we shall prove equivalent to the others. Definition 1.8 d. A set S will be said to be bounded if given any neigh- borhood U of the ori gin there is an integer implies ex. S e U. the statement of definition 1.8 b. ;It' As usual the letter T "JI::= -Y ( U} such that This statement evidently implies •ro prove the converse let S be bounded denotes the set of all elements of the linear topolo[~ ical space ;/, and 'T'..:.. U is our notation for the complement of U. 17 1,#5 according to Neumann ' s definition 1.8 b, and let Ube any chosen neighborhood of 8 . By the continuity of o< JC (postulate (II b)) there is a real interval Io containing O , and a neighborhood V containing 0 such that such tha.t 1 0 V <= U /3 S B-y- hypothesis there exists a real 13 c:: V p Choose ""/(3 f I. so large that ()(. s - (o/r.i) ~ S for I°" I < 'l-v Then for la: I < 'I-,, Consequently S C:: I 0 V C u is bounded a ccording to 1. 8 d. In the sequel when we say that a set is bounded we shall mean according to any of the equivalent definitions 1. 8 a - 1. 8 d. In linear normed spaces this definition of boundedness reduces to the usual one, in which the norms of all the elements of the set are required to be less thm1 some fixed po si ti ve number f' . Next we consider compactness. Definition➔~ 1. 9. A set S of a linear to:pological S}1£i.ce will be called compact if every infinite subset of S infinite subset of 5 has a limit point . has a limit point in S If every v:e shall say that S is comnact in itself. It is well known tha t compact sets and bounded sets are identical in a finite dimensional Euclidean space. This is not the case for all linear to1Jological s paces , since the spheres of an infinite dimensional normed linear space ( for exa.m11le the spa ce [o, 1.] * of continuous functions on ) , though bounded, are necessarily not compact. Cf. for example Alexandroff and Hopf (1.) p. 84. However we l,#5 18 shall now prove the following theorem. Theorem 1.6. let :J be a linear topological space with "basis" T, and let Proof: S c:: Every compact set of!!. linear topological ~pace is bounded. 'T be compact. bounded. Assume, contrary to the theorem, that S is not Then by denying definition 1.8 a we may assert the existence of a sequence { X '- ! , Xe- E 5 , a null* sequence l «,. J of real numbers and a neighborhood U of the origin such that for every ~ there is a ),. > -Y with null sequence { {3c. ~> )(> € T 71. 1 In other words there is a J and a sequence i -'(" J , 1'- E 3 ~" ~,_ e ,,, ..:... U t . .:. . U for all '- • has a limit point x . such that Now S is compact so that the sequence By postulate en b) there exists a real Io containing O and a neighborhood u.K with Io u,.. c U. Since x is a limit point of f .} there is an infinite sub-sequence 6 interval f 'f '" J with so that 6. fl"" €- I ~"t< 0 ~ Uk we have r3 .. " for all 1< • If we choose /< large enough "f',.,. e, L UJr c:::: U which is a contradiction. Continuous isomorphism; relations between linear topological spaces and metric spaces. What is the relation between linear topological spaces and other linear spaces which have been more extensively studied, such as linear normed spaces, Euclidean spaces, Frechet {F) spaces? Are there simple topological conditions under which a linear topological space reduces to one of these more usual types of spaces? We shall discuss these questions in this section, making our ideas precise by using the concepts of linear transformation and continuous isomorphism. * We call f«:" J a ~ sequence if 1,#6 19 Let d, , c/2. be linear topological spaces whose "bases" are respectively T, and 7;_ . A function or transformation (we shall use the words interchangeably} with domain T, and range ~ is called additive T. if JC, 1- l ~ € 7; implies -f-( >c + 1f-) = f (.x) -t- f (f) , homogeneous if ex real implies 1 f-(«,;c) = o< f(.x) is additive and continuous is called linear. . A function which By using approximating sequences of rationals it follows exactly as in the case of linear normed spaces* that any linear function is homogeneous, and thus is distributive (i.e. both additive and homogeneous). 11he space J, will be said to be continuously isomorphic to the space c:;( if there is a linear transformation 1- = f (x) with domain 7; and range Yz_ which has a unique linear inverse X = f -,(f) . It is clear that the relation of continuous iso- morphism is reflexive, symmetric and transitive, and thus is an equivalence relation for the class of linear topological spaces. This equivalence relation is a generalization of that of definition 1.4. Definition 1.10. A linear topological space will be called normable if it is continuously isomorphic to a linear normed space. By a me t ric space we mean a space M for which corresponding to each pair J<,jEM there is a non-negative number (X.>f) properties (i) ( x, f) = (1';,X) (ii) (;<, t)= 0 implies )( = (iii) (X, f ) + ("4'_,Z-) ~ (x, z.). * See Banach (1.) p. 36. f with the 1, #6 20 Every metric space is* a regular Hausdorff space if we take the spheres as the nei ghborhoods a Hausdorff topology . LJ~ . The metric is said to induce A linear topological space whose topology is thus induced by a metric will be called a linear metric space**. Definition 1.11. A linear topological space will be called metrizable if it is continuously isomorphic to a linear metric space. It is evident that in a normable space it is possible to define a norm by taking // ¥ II as the norm of the isomorph .x' of .x . over, since by definition the isomorphism X '= f (,c) More- is bicontinuous and bi uni vocal and takes the origin 6 into the origin 8' it is easy to verify that the »spheres•t for the space. II X II topology. ll X II ~ ~ form a complete nei ghborhood system Hence in the terminology of definition 1.4, the norm generates a topology equivalent to the original Hausdorff Conversely if it is possible to define a norm ind which gives :J then ::J is normable rise to a topology equivalent to the topology of according to definition 1.10. A similar necessary and sufficient condition for metrizability may be stated as follows: A linear topological space is metrizable if and only if it is a linear metric space under a topology equivalent to the or t ginal Hausdorff topology. The following fundamental result is proved in Kolmogoroff (1.). Theorem 1.7. (Kolmogoroff's normability theorem) A linear topological * See Alexandroff and Hopf (1.) pp. 38, 68. ** Our ttlinear metric spaces" are less general than the "vector metric spaces" recently considered by Adams, Trans. Am. Math. Soc. 40 (1936) pp. 421-438 since we require the operations of vector addition and scalar multiplication to be continuous. 1, #6 21 space 1! normable if and only if there exists a non-vacuous sub-set of lli space which is convex, open and bounded. I f we call a space locally bounded if it contains a bounded open set we have the Corollary The necessary and sufficient condition that a linear topologi-------- -- --- --- ----- --- - - - - --------- cal space be normable is that it be locally convex and locally bounded. We next consider the effect of requiring merely local boundedness. Theorem 1.8. If~ linear topological space :J is locally bounded then.:!!::_ may define ~ "pseudo-norm"* I)( I (a) f J< I ~ 0 (b) f«,c( (c) = , f KI= 0 in ;J with the following properties implies ~ = 6. IGC.I IXI Thespheres IJCI<'/,,, -Y=I,~,--- form~ complete neighborhood system of the origin. Proof: Let G- be a bounded open set containing the origin, and put V = G- ·(-G). Define l JI. I as I= , IX 0 Then V is obviously bounded, open, and in£ J«-1 5 k •· l•lJ then ,c 6- ~ c. X E « V . Then I JC I ~ 0 ; 11 If • V for each oe. .. of a sequence -fevior limit-. V= - V. f °'-c.. J converging to zero, by the definition of e;Pea1i ee1i loweF "eettrt4. Now since V is bounded, for any neighborhood U of the origin there is a ,, such that < 7 1> implies CX:c. Vc U (by def. 1.8d) , i.e. the sequence foe: .. V} is a complete neighborhood system. nei ghborhood of 0 , and )( Hence if = a . This proves (a). that I« x I - I J< I = O ,,,,f. / (31, x E- P/o,: V - , )( is in every To prove (b), note , oc. I ( ,·,,,f I fl/«;( :> X E P/ot; V) lcx.ffxl. * See Appendix II for a discussion of pseudo-normed spaces. 1,#6 22 Now consider the set all 1.2. )( € «- V., such that X This by definition is the set of I« I .c 'I-µ and this set is open by theorem U of the origin But by definition 1.8 d, for any neigbborhood there is an integer -P = -,, ( U) Hence the sphere f ,c I < 1 /-,,, such that ("' / <. 1/ p is contained in implies ~ V c:: U. U for large enough -,,, , and (c) is demonstrated. Corollary: If J is locally bounded, ~en ~ set S <= 'T' ~ bounded . if and only if there ~ ~ f', ~ O such that )( E S implies J XI < I"'. On the basts of this theorem and the results of the preceding section on compact sets we shall now investigate the relation between linear topological spaces and Euclidean spaces. Theorem 1.9. ~ necessary and sufficient condition for~ linear topolo- gical space ;J to be the continuous isomorph of.!: finite dimensional Euclidean space is that J Proof: contain ~ non-vacuous compact open set. The necessity is immediate since openness and compactness are invariant under continuous isomorphism (in . fact under any homeomorphism). To prove the sufficiency , let origin. By theorem 1.6 of section 5 G is bounded. a pseudo-norm Ix I Let G. be a compact open set containing the 5 cT Hence we may define for :J with the properties described in theorem 1.8. (where T as usual denotes the basis of ;J ) be bounded. 41'5 and therefore4is compact. Then for some Consequently 5 is compact, and the space satisfies a Weierstrass -Bol zano theorem. Moreover, since the first countability axion is evidently satisfied (the countable family J JC I < 1 /-,,, being a complete neighborhood system of 6 ) 1, # 6 23 every limit point is the limit of a convergent sequence. Hence by our "Weierstrass-Bolzano theorem" every bounded sequence contains a convergent sub-sequence. Following a method of proof due to F. Riesz, !~e,ca ~-=~·e:. {cf. Riesz. (i.)). we shall prove by a contradiction that the space is finite dimensional. If a linear topological space is locally bounded and L is a Lemma. which ;s, a. f>.,.Of~' sub.set o~ o. 1:"e ,u·· MO.tti~old M closed linear manifold,\ there is an K 0 f M of the space, then corresponding to every 11 .,. o and such that / /f.o I = .f. I JC - ~ o I > .1. - 11 L . for all JC Proof: Talce Jt 'E £- M .:. L Then since L is closed, and since the topology according to the pseudo-norm is equivalent to the original topology of the space, if follows that there is a positive number (.J' such that for all /3 1 • such nwnbers Given any Put Xo = (i.e. o~ > O )( I - .., £ Let /3 be the least upper bound of L /3 is the minimum "distance" of XE M there is a -f "" ILL from ;c' ). (3 ~ I)( ·---f I < (3 + a such that I Then f .(ol = 1 ~ IX '- 1- I I IX - X0 I K . Also - also, so that = f '+ /,c'-1'/X E L I lx'-:z.l ~ _p_ ;,,,,- rI 1' - ~o I - •x~1•1 Now if ;c € L, z I.IC~-J'I ra+ ~ r.rke lermna follows from this inequality. Now assume that the space 3 be any element with Io' I = .i is not finite dimensional, let 1/'• and let L, be the linear manifold °' -'f1. . By the lemma there is a ~ ,. f 'T' such that I '121-= 4. 1, #6 24 11'"• - '1"., \ ;, 1/2- . Let L2. be the linear manifold determined by 1 ., "f:1. . Then there is a .,.,,, E 1' with L"t, / = .i , 1-t.. -,; l ~ •1~ 3 for '- = t ~ ~ • ~ By induct ion we can determine for ~ <-,,, such that I q>'I =J.,, , and since we are supposing 'l' to be infinite dimensi onal , each finite dimensional linear manifold L.,, is a proper subset of T , and we can define an infinite sequence i. ,r,, l for with the properties f"'-,. ;> • I 'i" I= .i Thus the sequence { , I "#'IA- -..er-,, I > •/~ "'fc.J is bounded ( see corollary to tbrn. 1.8) and yet contains no convergent sub-sequence, which is a contradiction. We have now demonstrated finite dimensionality. B:µ.t in Tychonoff (1.) p. 769, it is shown that every finite dimensional linear topological space is continuously isomorphic to a Euclidean space, and the proof of the theorem is complete. Corollary. A locally compact linear topological space is finite dimensional. We next consider the questionmf metrizability. An important theorem due to G. Birkhoff* states that a Hausdorff group, that is a group with a Hausdorff topology in which the group operation and its inverse are continuous, is metrizable if and only if it satisfies the first countability axion, i.e. if there is a countable complete nei ghborhood system of each point. Now we are interested in linear topological spaces, which are Abelian Hausdorff groups under addition, and we shall find the following corollary of Birkhoff's theorem useful. (Although we write the * Birkhof'f (1.) 1,#6 25 group operation+, we do not need to suppose commutativity in proving the next t heorem.) Theorem 1.10. A Hausdorff group satisfying the first countability axiom is metrizable by means of~ metric Proof: ()(,j) satisfying (x-1r~ {I)= (x,-1") Birkhoff shows that by choosing a suitable countable complete ! V J of the unit element 0 , and defining an neighborhood system K "ecart" l> (JC, A ;, a ) = j ff f.r-"'t" V,, ( •~) K it is possible to obtain a metric for the space such that the metric topology is equivalent to the original Hausdorff topology, the metric being given by It is evident that Hence By combining the results of theorems 1.8 and 1.10 we obtain Theorem 1.11. Let :J be a linear topological space with the basis 'T' , and --- - - - --- --- ---"---------- suppose that.;/ ~ locally bounded. Then it is possible to define in 'r - - - =------- 1, #6 26 ( i) (ii) A "pseudo-norm" f XI ~ with the properties ~ (1.) I xl (2.) t«xf =L«-llXI ( 3. ) If I JC I ➔ 0 o IXI = 0 implies X = e , l -'f I ➔ 0 then Ix + f 1➔0 k metric (JC>'t) with the properties (a.) (..t...,"t) ~ 0; (b • ) (JC, 'f) (x.,1)=0 implies x =-t = (-t X ) > (c.) (x, Aj) + (1f',z) ~ (JC_,z) (d.) (.x --1'", 6) = (,c.,'(f"). Moreover the topologies induced EI_ l x I ~ original topology of Proof: cJ . If (;. is a bounded open set containing 8 the sets form a countable complete neighborhood system of the origin by definition 1.8 d, and (ii) follows from theorem 1.10. Property (3.) of the pseudo- norm follows immediately from theorem 1.8 when we remember that the function X + 1- is continuous at the origin. We are now in a position to discuss the relation between linear topological spaces and an important class of spaces known as Frechet (f J spaces. Definition* 1.12. A space E will be called a space of type (F) satisfies the following postulates (1.) E is linear ( 2.) £ is a complete metric space. ( 3.) The metric satisfies (x,f) * Cf. for example Banach (1.) p. 35. - (J<-"t, B) if it 1,#6 27 ( 4.) ... implies lim «7'# X = 8 lim «.,, = 0 for each X ~ E .,, implies lim ,, . _ JC-,, = e, ,, ➔ oo ( 5.) for all real oc, er;( Our main difficulty will be in showing that a space of type (F) is a linear topological space. For, while it is an immediate result of the postulates that Eis a Hausdorff space (being metric) in which the operation X-+-,t is continuous in X and1 simultaneously, and in which «~ is continuous on the left, and also on the right (cf. Banach (1.) p. 35 for the details), it is not at all clear that «X is a continuous function of both variables simultaneously. However we shall now prove that this is indeed the case. Theorem 1.12. A space £ of type (F) is ~ linear topological space, that is «. JC is ~ continuous function of oc and X simul taneous1,. Proof*: It will be sufficient to prove continuity at «: = 0 , J< = 8 since we have (°'~~oc 0 X0 ) ~ ((«-« )(x-, 0 0 ), 6) + («0 (,JC-JCo)Jli) -t-(fa1-~4)Xo,8) . We have then to demonstrate that corresponding to any positive number there is a positive number ~ such that (ot.X✓ l>) "- ""i . Let for fo{ I~ S contains 8 and hence is not vacuous. H,, and H,.. denote the set of points of E such that Clearly each JC,. E "t'\ Also each . H-,., is closed. be a sequence tending to a limit oc. satisfying foe..' < 'l-y 't. we have («x,.>e) ~ 11 • H,. For let Then for any chosen :ror . ~1 C, • * The method of proof was suggested to me by reading Deane Montgomery's paper, ttContinui ty in Topological Groups", Bull. Am. Math. Soc., vol. 42 (1936), pp. 879-882. 1,#6 28 But since o< X is continuous on the right, and since (;c,1) is a contin- uous functional of )( for 1 ::::. 8 ( at' t., (J) ~ we have 11 • , that is H~ is closed. Since « X is also continuous on the left, every point of £ is contained in some H.,, . That is E =~ s---1 H,, But since the space £ ( the union of the H., /s ) . is a set of the second category*, there is at least H ,s, say H,.. of the second category. H,,. is everywhere de nse** in an open subset G of £ . But H,,. is closed, so that G c H,c. . one of the Let Xo be any point of such that of G . Then since G is open there is a (x., Xo) < S, G c H,.... and l .... .L.. 'la.c. r implies that JC E H~ we have that ( ~, l<o) L.. c•• Let $ be the smaller of $, I c,(. I <. ~ we have J \ Theorem 1.13. and . imply Then if ("f , 9) < (« 1 , 8) = (« ( 1 + Xo) -oo<o., 0) (J< .J Xo} < $ , $, > 0 By the definition 1 / fk . and ~ (oC("t-t-Xo).,a) -t-(oelCo.,6) ~ :l"'f, < 1'f (1 +- )(o, Xo ) < $ , Hence ( of J<.J 0) '-~ <M·• • S and since obviously . This proves the theorem. sequc•tit1ll'J A necessary and sufficient condition that ~ ~complete linear topological space ;/ be continuously isomorphic to ~ space o f ~ (F) ~ that :.J satisfy the first countability axiom. This theorem is an immediate consequence of theorems 1.10 and 1.12. * A set S of a metric space M is said to be of the second category if it cannot be given as the union of a denumerable number of non- dense s ets. See Banach (1.) p. 13. ** See Banach (1.) p. 13 , theorem 1. 1, #7 7. 29 Realizations of linear topolog~cal spaces. The following instances indicate something of the scope of the theory, show the independence of various properties of linear topological spaces and serve as illustrations of the preceding theorems. None of these spaces are normable. Example 1 °. Fo~ the linear space T dimensional vectors where each X c.. ) x +,t = (x, + o/,, x.,_ +t,.., --- ) real number, trt:¥,.l~ )( = (it,.,;<,., --- (Gt'.(.,«,(~ f or- lrJ.1) As neighborhoods of • , is a and defined as the set of all The elass V of all infinite take the set X we take the sets 1 satisfying (.. = 1,:z., --- J ,,_ of nei ghborhoods of X is obtained by; letting "7' vary over the positive integers while~ varies independe ntly over the positive real numbers. ?J- , and we shall now We call the space thus topologizec. show that 1J satisfies all the postulates I - IV. ( I a) Obvious . (I b) Given U(x ; >', S} and U(x; f".,, 'f) it is clear that U (x; .>.., ~) c (/(x; ~ ~) • U ( Jf~• fk.1 ?! ) 'f {I c) (. If ,.~,,· ~ ht i r'b I 1 A < j, ,- -1' f U(x; -P~S) then If JC + { II b) U(x;--YJ~) i -11', - J<,I., - - ·., S- I1f--,,-K.,. I• t , then there is an integer -Y with I 1<,, - -"I',. I > O Obviously {II a) , ,-A ~ "'} . U(1 ; -Y~ 11) c providing ~ < the smaller of ( I d) providing ~ > t"., ...4 > ".,, U(x; Y.J Y,.[,r,,- "f,,I) does not contain f. Obvious from the continuity of each component . " tt I! II " tt II 1, #7 30 ( III) (Local convexity) If ~ « I x .. I + (' -«JI y .. \ .c. JC~ 1 ~ ()(•; -, ,~ i) then J(t'l<a. t-(• -ot)-f•/ a for o <- « 4 / • That is each U(6; -,,_.S) is convex. ( IV) ( Completeness) out of ,,._ V . Let f x J be a fundamental sequence o.f vectors 0 '' l ".. o> - X rr)I ~ tends to zero for each L as >-. and g tend independently to infinity. But by the Cauchy Then it is clear that y' necessary and sufficient condition~ for convergence of a sequence of reals, th ere i s, for eac h C. , as ).. becomes infinite. "' such that an ,.~ I x ~>> ... - x._ \ tends to zero But since neighborhoods are defined in terms of r a finite number of components, it follows that the sequence 1. X converges to X , where ,,.>? } ;< = ( x,, x~J - - - ) . Thus the space 1J- is a locally convex, sequentially complete linear topological space. Moreover, since bility axion it is metrizable. V obviously satisfies the firs t counta- (See theorem 1.10). In fact if we examine the well known* metric I we shall find that the metric: topology is equivalent to the above neighborhood topology. To prove this let a. be any point in V and consider the where o ~o<L I together with the neighborhoods and o .:. a < + O'O. * Cf. t or example Banach (1) p. 10. extensively by Frechet. This metric for V has been used 1,#7 31 Givan U (a.; -,.,., ~ J put «: - __i;;..__ - 2 .,.(, +$) 1 ICe, - a."'1 < - ~implies ----....;..,:.c I .... IJ<c. - a,,. l l + • or '.l'hen ( x, "') <.. D(. a,., I < 0c. ( x,_ - That is, K(a,;c:ic.J C be given. Choose .,, so large that _.z:! ~ U(0-;,--,,_,$J) . '=~;,,~ ---:., "'· , on the other hand, let K(4';«) .., , and choose Then if I¥., - a,~ I < S for we have z ~=... I 2>- IX,. - (1.,- \ I + 1¥,- - «"' I + That is, . two topologies are equivalent. oc. ;2.., Hence by definition 1.4 the With this metric 'V1s a space of type (F} , and thus illustrates theorem 1.13. On the other hand 7J- is not normable since it is evident that no neighborhood is bounded. Example 2°. For the linear space T take the space F of all real functions f(V of a real variable defined on - _,, ~ \ < -t- 00 , with addition of functions and multiplication by reals taken in the ordinary way. ~ eq'el:ality of fun.ct ions takeB: ae oqu&l.ity ef t:ao v:alQos ev:0Pjt11,th~re. ponding to any finite set ( i ,> - --., \.:,,) CDrres- of real numbers and any 5 7 ° U (f; t,, ---., f,,, S) of the point f. for which l <j ( \ r.) - f (j'" JI -<. ~., L =- t_.;z.J - ·: '1'. we may define a neighborhood as the set of \>oints ,, j We may verify by methods similar to tbose used for example 1° that the space -:/- with basis f and topologized by means of the neighborhoods is a locally convex, sequentially complete linear topological space. l, //'7 32 The space =f- provides us with an example of a linear topological space which is in fact locally convex, but which is not a Neumann space l. , because Neumann's postulate (2.) requiring the existence of a countable set of neighborhoods whose intersection is the ori gin is not satisfied. To prove this, let l U(e~ J;.J ---., J;.,., ic.)]be any countable sequence of neighborhoods of the origin 8 ( e is the identically vanishing function). Now the union of the countable family of finite sets ( }~.. - - ·, j ~") ( },_., ~ ..... ts., - - - is obviously a countable set, say ) But since the real numbers are uncountable, there is a real number . f .9 ()) is contained in each Hence the function 3(l) defined by = 0 for j (to)= i U (e; j ~, - -- }:") but does not vanish identi - cally, and Neumann's second postulate does not hold for the space =f- . It is now obvious that the first countability axiom cannot be satisfied by 1- , so that the space 1 , unlike the space 1f , is not metrizable. Example 3o (Tychonoff) As a basis ta.~e the linear space infinite dimensional vectors oo I X Hva of all = (I<,, x~.. - - - ) 'I'$. the series ~ I X c. I ~ =' Now define a nei ghborhood converges, and put U (J<~ol,) as the set of points 1- for which (1.) (2.) There is a positive S = a <-tJ such that l z.. -1- J -< ~ implies Clearly, for all oc. "?' o , ;< E U (x; «-) For the proof that this space is a linear topological space see Appendix II, where the general 1,#7 33 class of "pseudo-normedtt spaces is discussed. Thi s example is due to Tychonoff (cf. Tychonoff (1.)) who has shown that the space is not locally convex. Thus an independence example is provided . for the convex- ity postulate (III). In addition, this example taken toge ther with say, example 1°, shows the independence of Kolmogoroff's two conditions for nor mability , namely local convexity and local boundedtfes$. For it is clear from definition 1.8 a that every LJ(x;o<) Example 4° (Neumann) is bounded. As a basis take the real Hilbert space f = (x.o, )< ... - - - ) the set of all infinite dimensional vectors 00 which the series ~ l )( c. \ ,:o #,~ ~ { f .. ~ ) = neighborhoods C. t C , H,._ that is for ,. converges. As is customary we put where . U( -f-0 ~ <f},, - ·:, c().,., $) Define as the se t s of points 'J The set of neighborhoods of f-0 is obtained by letting c;\>,, - - - ., <f),. sets of vectors in vary over all finite H~ and letting $ range from O to +oo . In Neumann (1. ) and (2.) it is shown that this s pace ~w with the basis H.a- and the above "weak" topology is a locally convex, sequentially complete linear topological space which is not metrizable , and hence not normable. We may also prove that fl/.,., is not normable by showing that no neighborhood is bounded, as follows. Given any finite set cf',.., · · ·.. d{,, there is a vector 410 + 0 for t.=1, 1 1-·-y . for any chosen orthogonal to each c:fJc. Put <j..c.:;co£cf)o-t-fo Consider the neighborhood U ( 9 ~ {)o ., i) , that is (cfo, f .. ) = 0 Then 'J« f s; , and for all real oc , since of vectors, U(-fo;1,.-··.Jcf>J~) ( ~ - .fo, ~") = °' ( fo., <(1...) = 0 Then no matter what pf.o we * An alternate proof will be given in the discussion of example 5°. 1,#? 34 choose, there is a 'JCIC. (3 9« i such that evident from the equality ( f '"' , <(} 0 ) (q)o)<Po)~o. and the fact that U (e ; cf>o ., 1) as is = f3 a( ( cf)d.> cf>o} + {3 (fo-> <(Jo) Hence by definition 1.8 b, is an unbounded set. Example 5° (Normed space with weak topology) Let {;, be a linear normed space with basis £ . Besides the ordinary "strong" topology furnished by the norm (cf. last paragraph of #- 1) we can define a "weak: 11 topology for the space in terms of the linear functionals on £ . Denote by real numbers. elements of C. 1 on £ £, the set of all linear functions -f(;c) Corresponding to any finite set (f,,, f~. - --~f.,,) , S >O any , and any x f- £ define to the of (J(;r~-f,.,··;l,,.,S) as the set Then we can easily show that the space c;w with . basis £ and with the topology furnishe d by the neighborhoods U(x; f,.,•-~f..,J fJ} is a locally convex linear topological space. ( I a) Obvious. (I b) Given Ux = C '~ Take o we have ( I o) If U(x;i-,,---,f.,..,r.), • ~ ~"' c C' o., ii • Then - r 1T u; = U(x;9,.,- --,,,.,,$') U,I(,/ =- ()(.x .}- f.'• ··· f a --· ,,,..> b ,,),/ I ~) -JI .I .I U; C Ux • U; . "f E LJ (;r;f,.---..,f.,.., $) , i.e. put S,-c. = ..c ,·~ for <- = I,···., -,, ' Then for all z. satisfying • •~C.i 1" ( ~ - If.. (1 -x) I-<-~ l-fc.Cf-~) I) \ ft.(z-1-J \ < 51' 1, #7 35 we have, since ~ each f 1. is linear, that l-f-"Cz.-1-JI -r \-fc.(1--xJI + $.,_. I+ .. c4 - x >l <- ~ That is (I d) Suppose JC :P ~ 11-f-XlJ >O , so that . By theorem 3, p. 55 of Banach (1.) there exists a linear functional f, (z.J such th.at Ta C1- -~) = ll 1 - ~II . (II a) Clearly U(x-t-'lj f,~---.,f,. . . Given on £ 1-·€ U (;c ~ f,, 111-JC 11 ). ~J , note that if l f" (JC '+-1' - X- 1) ( = l -f.._ (x' -If} + f-v ( 0 '-d') \ < j/z. -+ Sf;t :. J> That is U (JC; -f-,. · • ", .f ~., J/-i) -t- LJ (1- ~-f,, - ·-.,+'7'., S/2.) C LJ (.,c +-0 ; f,.,. - -.-f,,~ J) ( II b) and U (.l JC~ -f,, • ··1 -f_,,-> S) , choose positive numbers ~. Given a.. so as to sat isfy the inequalities } a:a. ( s, + I f.,(.-¥Jl) < 1J/2. .) (_ lot. l a, j/:L < Then if I« denote the real open interval all 1'~ Ioc. and all f satisfying I I 'i' - Q{. I~ $ .. \·\\.Cd"-;<)\"-~. , we have for for c... == 1.,;1._. -··,-,, that li ... (1' ~ - t1< x) I - l-fc.(C~-"')"Y -ro{(-1--x>)\ ~ 1~-«1 t-f,c 0 JI 1- 1--tl 1-f.. <1 -xJI < sl. ( s, + ff'" (JC) I) -r l~ ' s, < 36 1, #7 or, in the notation of the calculus of sets, C: U ~of)( • f. -.. -f \.<1111 J • ., ., ., , ~) • 2,#1 37 Chapter 2 Integrals in_!:, Linear Topological Space 1. Introduction A considerable number of authors, starting with Radon (1913) and Frechet* (1915), have studied integrals for functions of which either the domain or the range are suitably restricted abstract spaces. In 1935 G. Birkhoff (Cf. Birkhoff (2)) developed a very general integral of the Lebesgue type for function.a, with values in a Banach space, basing his work on Frechet's integral for real valued functions on an abstract "measure" aomain. In defining his integral Birkhoff does not need to suppose at any time that the domain over which the integration is to extend is of finite measure, or that the function to be integrated is bounded on this domain. The first four sections of this chapter are devoted to an extension of Birkhoff's integral to functions with values in a linear topological space, following his suggestion of (Birkhoff (2.), p. 377 ) that his integral might be extended to linear spaces** more general than Banach spaces. We continually make use of (1.) the idea of unconditional convergence of series, due to Orlicz in the case of linear normed spaces, (2.) the calculus of' convex sets, (3) the properties of convex sets. * Cf. Radon (1.), Frechet (1.) ** After I had obtained the results of this chapter, Birkhoff (3) was published, in which there is contained a definition of a "Lebesg ue" integral for linear topological spaces. This definition does not coincide with mine. 2, #1 38 Besides the properties of convex sets given in Chapter l, #1 we shall need that provided by the following Theorem Proof: If S is convex so is its closure. For x, ,:,- E .S for o < °' < 1 we must show that Given any Uz , by the continuity of vector addition ana. scalar multiplication there exist U~, U1 , 1 -1' E Uz. Therefore L :z'E S and since Uz. is arbitrary, z :: such that x'E Ux., 1'~ U,- • there exist J< ', -t' € S Since ¥, 1 E S «.)( = « x -t- c, -•c)-'t' E S z + (1-cic.) But since is convex, ES. From this theorem it is evident that closed convex set containing S . S Gr 5 is the smallest In this chapter it is convenient to denote a linear topological space and its "basis" by the same letter as we shall have no occasion to consider spaces with the same basis but with different topologies. ·Throughout this chapter T will stand for~ u v.w--- locally convex, sequentially complete linear topological space. will denote convex neighborhoods of the origin 2. J ., e . The domain of the independent variable. We shall be concerned with functions F(~) on a space* £ to 't, where £ satisfies t he following postulates: M 1. There exists a family ;/) of "measurable" point sets A of E with M 2. The logical sum of a finite or denumerable number of sets A f ~ is a member of lJ . * These postulates are a modification of those used by Birkhoff, in which the family ~. is not explicitly mentioned. J 2, #2 39 M 3. If 4 f2:J M 4. There exists a sub-family 2J, so is its complement E-=- ~. of 2J such that to every .tl of 1'.>, there corresponds a non-negative real number At (4) , called the measure of M 5. 4 . If A =.;ti Ll,. joint sets Aa. is the finite or denumerable logical sum of dis- of 2), and if the series ...::Ef A« (A .. ) then 4 E i,. and MC ( 4) =..,,Z: .-t- (Ac.) . then M ?. Every set A of 2J converges, A' E ~ 1 . can be given as a finite or denumerable logi- cal sum of sets .A, of 1:J, . The consistency of the postulates MI -M 1 is shown by interpreting £ to be the set of all real numbers, 1J to be the set of all Lebesj ue measurable linear pointt sets, including sets of infinite measure, and ~, to be the set of all linear point, sets of finite measure. From M 2 and M 3 it follows by taking complements that the logical product of a finite or denumerable number of sets A c. f X, If £':I= 0 is a member of ~ • is a member of 2) , then C' can be made to satisfy postulates M l - M? by a process of relativisation. , (read the set o;t all , Simply put c.;'\ E ·A s for which 6 f .J and and M 1 - M? are evidently satisfied by E,. 2,#2 40 Z> Since the logical product and the complement of sets of are sets of ci© , it follows that the logical difference L\ .:.. .6 of ~ and A Ll f rJ 1 1 • ( E ~.ct) A - is in 2> f'or A, A'€-~ ~ =~ ..a L that , . Now, from M 7 we have for where .Ac. A=~(~'--=- (.Z: A,.)) = ~ A: M 6. = Ac. ..:.. ..Z:: A,. >#&. If A =I= 0 and obtain A are disjoint ' XJ, . Hence , where the sets J.#1, A./'- E and belong to~, by M 2 and , then we can reject all the null sets Ac. as the sum of a denumerable number of non-vacuous dis- joint sets 4' '- E jl, This result is fundamental for the theory of integrals in #4. Unconditional convergence of series in T. 3. A series .Z:: ""' = X, +;<z. -t-1(3 -t -- - , ff' X1. E will be called unconditionally convergent to an element J< E' 'T corresponding to each U there is an integer s -)( f u , where 5 is any finite sum the series and including -Y> 0 if such that .Z:: X >-~ taken from ~ In general it will be convenient to denote infinite series, and also their sums, when convergent, b y ~ }('- , and finite sums, each of whose terms is selected from an infinite series by _.z: Xe.~ . ~ ,~:/x,., A necessary and sufficient condition that !:. series .. Theorem 2.1. be unconditionally convergent _!! that corresponding to each U there _!! a 7/ have = ~ ( U) f J< L , such that for all finite~ in which ~ E U. tp > >" we 2 ,#3 41 Proof: To prove suffiency, let the condition of the theorem be satisfied. By sequential completeness the series converges to an element U V- I/ arbitrary take V .zj Xe., (: including x , , - • - , J< ~ ~ for elements not in S in S • u and there will exist ,., > -,, For with If .S is an arbitrary finite sum be the sum including all , but such that p-1 is less than the greatest index Clearly .S + ~ )(!t'-1 'J so that C K • - E V )( )( e.V-VcU . s - The necessity follows in a similar way. A countable set f X .. J of point sets out of T is called un- conditionally sUilllllable to the ~ .X.. if every series ~ J< ... , X,. t::X:. c. is unconditionally convergent, and if X is the set of sums JC of all =X these series . In t his case we wr i te ..Z X,. . Theorem 2.2. '!l1.e necessary mw. sufficient condition t.he.,t {.X .,] unconditionally summable _1!3 ~ corresponding 1Q every be U ther_Ji ~ l&. implies Proof: Sufficiency follows at once from theorem 2 .1. assume that £X-"J is unconditionally Stlllllllable but that the condition of the theorem does not hold. Then for some U and any 'Jl0 a set of elements X l'-t f X ,-,.~ ~ J< I"~ c:: ...:... S To prove necessity U such that ~~ ? 'Ya and This however implies that there is a series ~ J<,,_, .<pf.K,c,.which is not unconditionally convergent . diction proves the theorem. , there is This contra- The fol l owing theorems are fundamental to 2, #3 42 9ur theory of integration. l f ~ sequence or W§ Theorem 2.3 mable then f Co- X:.,, J is also unconditionally summable and • _.2: CrI" Proof: f X,.} is unconditionally rn- Ctr.z; X,, . c:= Let U be any chosen neighborhood of 0 . locally convex, there exists a Vc U such that By hypothesis there exists a -v0 such that Since T is regular and i, ~ 'Yo implies ...Z:: X,. c:: V. , 'j' Hence for any finite set of , ; s , t.,1 7 -Po , and the sequence { Co- X...,_ } is unconditionally summable. Since ...i: n . c: c.=1 -Y > o for all integers , it follows that in the limit Uncondi t ionally conve rgent double series may be de fined by requiring that every arrangement of the double series as a single aeries be unconditionally convergent. From the corresponding facts for single series it is clear that a double series convergent to an element X Z XJ.fA- is unconditionally if corresponding t o every U there is a ..,,(U) such that for any finite sum s out of the rectangular array sud, tluit s ~ contains ~ ,. ",/l=, X>.,-,. in theorem 2.1 that ~ only if for every X ,.,.,. we have S - .x E- U It follows as is unconditionally convergent if and U there is a ~ (U) where either all such that every finite sum A. 'I or all f'-r. are greater 2, //3 43 than V ( UJ . in case Z Since the space is regular it may easily be shown that x,.,,.. is unconditionally convergent to JC , then the "sum by rows" and the "sum by columns" both exist equal to x . 00 00 .Z f'.i. = p, .. ., ~ µ,..,. = f', ~" >=-• =, Theorem 2.4 If f",,._ "> o, .X.u cXc.., X ... >- c::x;, - 00 C. Z: /Le.Xe..= .x.., a.;, then is unconditionally ~ • fL,> X, > summable and ,,A:1 co ~ ff,,,.. ..x...,>. Proof: "·" =, Denote by X,. +- 8 c::: the logical sum of the set X~ 7 set consisting of the zero element 8 . Then since ~ ,l =• and the f'u < f"c. we have :, the sun being extended over any finite set of A. 1 S . and Hence s i ftf i ( 4'" I 'J Now ff'-',. A,. J by hypothesis. and Hence if'-a. JS.." -i- e J f ~;A;,. +-8} are also unconditionally J summable, by considering unconditional remainders, and by theorem 2.3 and 2,/13 44 are unconditionally summable. From (2.1) and (2.2 ) it follows that is unconditionally summable. Now c:: - by the definition of Co- X.... . Since the "sum by rows" of f f',c.>. X::,"" J exists equal to ~ f'-,.,.'X,_A we have in the limit c..,A=, C: or C where the last step follows from theorem 2.3. C: G,- .x: ', - and the theorem is proved. Theorem 2.5 If /"'-" - By symmetry ~O , ~ !'-,;,. = /A, c. ,A::, is unconditionally ~uromable and -?--: µ,,. X.,J. t._,A:. I Proof: C Cr X The proof differs from that of the preceding theorem only in 09 the method used for showing that ~ /-'-,Ji X .. ,. ,.,.1::, convergent. is unconditionally We have formula (2.1) exactly as before, and we can conclude 2, # 3 45 from it that given any U there is a "Ya such t hat i,& > ~ implies co r . z µ,,,.., >-, 'c~,., c: U · In order to prove that l fl'c>~A ~,~"'' is unconditionally surmnable it remains to show that corresponding to any U there is an integer J..c, such that 1 ~ <.., ~ 'Yo J and implies Let U be any chosen convex neighborhoi!>d of 0 the set .A.1. is bounded, there is an « .. > 0 {cf. definition 1.8 d). where f'-c1i >O . Since for each 1.. such that c:zLX:. c U From the hypothesis that~ -, l'-c.A converges, c.,A:, , it follows that there is a positive integer such that -v!J > At1 implies Hence, from the elementary properties of convex sets and the hypothesis we have <= Ctr ( ~o... XL + e) c::::: CD- ( ~O Consequently, if ),. 0 = Ima ;<>'o )\ " ~L~ that u) = ;du , we find for l ~ Lr,; ~ -Yo , .,,, ~ ,, .s "Hf"'" ,vis ) ,-______...,.,...___ ___ _ _ ___.... ;-...L ,,0 where we have again made use of the convexity of U ct, distributivity law on the left. u l = -1iLJcU J unconditionally convergent and the theorem is proved. ~ in applying the - This shows that ~ f,L,,~ .A.._,. ',-'=1 and is 2,#4 46 # 4. Lebesque Integrals. By a partition of the domain £ of the independent variable we shall mean a set TT , finite or denumerable, of disjoint non-vacuous sets Llc. € ~, Let whose point set sum ,.Z:: ..6, = £ f(J;) be a function which orders to each :,t of E one or more points of T denote by . If TT is any partition of E into sets ~, '=' ~ , ~ the set of function values F(A ... ) We shall say that the function F(-J;) 7T if the sequence of sets ..,..1ot,( (.a.c.) wi th J:; f= 4. '- F(A;) is summable under the partition F(.ac.) is unconditionally sum- F is summable under 1T and each set /H'f (A._) F (A,) is bounded, we shall say that f is boundedly summable under the parti- mable. tion If TT . Definition 2.1 Let TT be a partition under which f(;I;) dr- ~ A#l { Lie.) F(A,,) The set for A~ . Ac. F is called the integral range of relative to TT and will be denoted by that the set is summable. J,, (F) . To emphasize the faot belongs to TT we shall occasionally write We may now prove the following fundamental theorems on the integral range of a partition "consecutive" to a given partition. Theorem 2.6 F i_f TT, .!!!!_d TT,_ g_e fill¥ two partitions Jd £ i_s summable and if TT, • TT2 !..!J, denotes ~ vartition of E f A~ •A"'!:"'}, t heK. F ii !.!1B QJ the non-vacuous setsAsummable under TT, • IT-a. Proof: under ~ Note that theorem 2.4 applies here if we take and consist- 2,#4 4? =~(A":) J I f" J, x,. - f(6W:: )) = .,.....c ( A ' , ) , -,; -r,, f,,4 :: .,."'11 {A L • Ll :~) ' XL). "" = F ( A':• ·A~&) K~ - F ( ~~•) . is unconditionally convergent and is contained in both ..ti Cir .+t1 (A~•) F(A":..) " ,.Z: C,- ..,.ff'( (A•:_) F (A':'J . Consequently, on taking convex ;. and in hulls we obtain ~ _ II I • 1/J. Theorem 2.7 £. Let ( F) c.: J,, ( F) • J,, ( F) . I t-' : (A,.,Ll.a.., --), Ac.•d"=~ be!:. decomposition of into sets L11- of ~ ( Ac. not necessarily of finite measure) and let TT be ~ partition of Let TT• "P sets in then ,a. E. into sets denote the partition of ¼Li' c. • A,. } . If £ .A~ of ~, . consisting of the non-vacuous F(-J:) is boundedly summable under F(i) is also boundedly summable under TT • 'f' Proof: r,). and = /H-C ( 4 : • b. >.) J = FC 4~ . L1>-) ~ .x" = F( LI~J., 'l'heorem 2.5 applies here if we take ~, = .,NK rL<J,, TT jl- ,,. and we obtain _zj ,,H({Ll; · Ll,d F ( ~~ -Ll~) c:: ... ,.=, Taking closed convex hulls of both sides, we have 2,#4 48 Definition 2.2 A function F(J;) (single valued or not) on E 'r is said to be integrable over E to if there exists an element j(F} of T and a sequence of partitions TTl' with the property that for every U that J rr.,, ( FJ there corresponds an integer -V0 The element implies j ( F) c: U. - j ( r) Definition 2.2a such that -,., >Yo is called the integral of F(1:} If f(£J is integrable over £ over £. . , and if f is not only sunnnable, but boundedly summable over each of the above partitions TT.,, then we shall say that Theorem 2.8 is boundedly integrable over E . The integral of definition 2.2, when it exists, is unique. f -11-,, } Proof: Let 1. grals i(F') F and (,-r~j t , be sequences of partitions defining inte- and j(FJ respectively. Jrr.,,. rr; c: J rr.., • J,., l for any -,,~). . Let U be such that V- Vc Then by theorem 2.6 be any neighborhood of the origin, and let V U Then by definition 2 .2 we have for . that Jrr.,, ·tr' ). j(F) C: V i.,(F) c V J 11'-,, ,if,,_ + Hence . ,. this implies But since 8 E J,i: ·'tt' ~(F) -j(F) t{F")-J·(r)c: v-v. t v-v c::::: u, 2,#4 49 whence 1,. ( F} - ) ( F) i (F) - j (Fl , since T is a Hausdorff space, and is contained in every neighborhood of 8 definition 2.2 and the fact that T . From is both regular and locally convex we have the following theorem. A necessary and sufficient condition for F{~) to be Theorem 2.9 integrable ~ 'l"" is that there exist !! sequence of partitions TT~ of £ and ~ element j ( F} of 'T' such that, given U such that -,, > ~ there is a A :::.A (tJJ implies ~ ~{A.~) F(Ll,.J CJ (F) + u. Ac, ETT.,, Th,e following 11 Cauchy criterion" for the existence of an integral is often useful. Theorem 2.10 sufficient F(J;) be integrable il is necessary and In order that rn there exist .!, sequence of partitions Tf,, with the prop- erty that, given U there is ~ integer >-=). (U} such ~ 71 > >. implies Jrr.,. C: u. Proof: Necessity is obvious. let .,,0 be the integer corresponding to U in accordance with definition 2.2. To prove sufficiency let U be chosen and From theorem 2.6 the sequence of closed sets has the properties (1.) Each set contains the following (2.) JTf", ... tr,_ - y W, · · ·'11",- C. .J;r-, - J'ff",. C LJ 2, #4 50 By the axiom of choice (applied here to only a countable number of sets) there exists a sequence fx-,, J1 ;< -,., ~ sue h• that 1 u..-, .. ·TT~ . Then , so that by sequential completeness, the sequence f~,,.J converges to a point J< E 'T1 Each of the closed sets Jrr, ...;r>' for each -,, • Hence contains )( so we have X E ~ , . J...11",-, -x<=U , and F is integrable to j(FJ= JC Corollary: In theorem 2.10 ~ may replace - Jrr,, El. -2:! ~ (.o.,) F(4c.) ""~ TT.,., and still have a true statement . -- --- --- - ---- By comparison with G. Birkhoff ' s definition (cf. Birkhoff (2.) definitions land 3, pp. 366, 367 ) of an integral for Banach spaces, we see that our integral reduces to Birkhoff's in case '"r and the function F(*) is a Banach space is boundedly integrable in accordance with definition 2.2 a. Theorem 2.11 If F(*) ~ integrable~ E ll is integrable~ every set A of lJ to ~ element j (F~ 4 ) Proof: Let A,, € ':l:) I F(:t) be summable under a partition TT of E into sets and suppose Ll E ~ F( 6'.. ) C F (Ac.) F ( ~\. ) F (A") C: . Then 2, #4 where 51 A I ~(. - Therefore Evidently An (At) M1 (A~)(},. ~ (Ac.) <. so that P (A~) c:: /H'1 (A~) fo. ( F(A~) -i- 8) C4- ( .M< (~(#) F (Ac.) + 9) c f From theorem 2 . 2 the sequence A"'1 ( 4 c.) F {Ac.) + 0} is easily found to be unconditionally summable, since by hypothesis i s unco~~itionally summable, and hence by theorem 2.3 we may conclude that the sequence of sets ,,m ( A~) f:4- F{A~) = (i MtO:i:) F(Ac.) is unconditionally summable. .ff.,- A'l"'f {A~) F(A~)3 A similar agrument shows that unconditionally summable. and the fact that M -t- N is From the distributivity of the ··e.o,. ,, operation c:::. M -r N for any two sets MJ N in T we obtain c::: That is, ( 2.3) Jrr, ( F~ Ll) + where J.,,, ( F~ .6) J1T1. ( F~ E .!. A) c::: Jn (F,, E), for instance is the integral range of F L1 with respect to the partition Tr': f Ac. •A J of A . over Now consider 2,#4 52 a sequence TT.,, of partitions of £ for whi ch J-,,,, - J-:~ 0 11,, in accordance with theorem 2.10. Given U we have for all -,,;,A(U) on using the inequality (2.J) for TT= TT.,, , ,Ir,;_ (t-J A) + J,r;, (r.., E...:.. Ll) - ~~ ( F;Ll) - J _ ( F, E.) - J rr. ( ~ E) c: U . c::: " .,, .,, c[,,~ (F,,A) - ~•,,( f:, 6) ~ ,._ (F e.:.A) - T (s E.;..tJ) each contain 0 1i' '1f ~ But since the sets and , this implies that ,I 'JI J,.,,, ( F., E) and for all -y > A(U) grable over 4 . J._._ (,=-., £-A) u C: It follows (theorem 2.10) that to some j ( r.., A J and over t:. -=-~ F(~) is inteto some J· (~E .:. A) . As in the proof of theorem 2 . 10 we have j ( F; A) -r- j ( F., E ~A) c: d-rr'.,, ( ~ Ll} + c for ""JI > A ( U) ., '¼~ (F:, E ..:..£j) J v.,, (F-> E) c: i(F;EJ +U for arbitrary U . Hence, -by the wai~of¼eee tl:lee:Pem The theorem now follows, if we replace £ by the "sub-space" E'. Theorem 2.12 function Proof: If j (F,,A) r(.tj is boundedly integrable ~ E then the set is completely additive. Since any member £' of 2J by relativisation satisfies the postu- lates for an M-space, it will be sufficient to show that if '"fl is any 2, #4 53 de numerable collection of non- vacuous s e ts then 4£ E 2) converges and is equal to To prove this, let IT.,,:(--·..,Ll~ .. --·) with j (~ E.) , where ..1: ~ '£) , be a sequence of parti tiona of E such that Having chose n U let -,,0 be such that ..,... > Yo . '1T.,, • '"P -v > Yo , for any . implies Consider the partition By hypothesis each set F(A ~ ) is bounded, so we can apply theorem 2 . 5, putting -,-,-:,. IT,...J ~ .. ;. = ;ff(.(A"':,-Ll.,- } ., .X-,~ = ~ .. ==..+vr ( At).., F ( Ll...~ • ~A J :, .X. c.. = F ( A:), iMf (A~ -~.d F(~ -: ·A,.) J and conclude that is unconditionally summable and that the following inequality. is satisfied. .z "~~=, O& ( 2.-t) .MC ( A~ · Ll>,.) r (A~ · Ll~ ) c::: GO (r _2J /W1 (~':,) F(A ~) c.=, J7T ( 1--> E) . A>,. , we have -,> Since by theorem 2.11 F""(J;J is integrable over 00 j (f"., L\,-) C J;,,.-A,A (FJ4,.) = &~ ,.m(4~ -L1)1) F{Ll~-4,a) Swmning on ~ from 1 to oo , we have Ct,. zc.=, ,M,<(A~ -AA ) P(4~ · ~~) 2,#4 54 But from theorem 2.3 this implies that ~ -=--=------ - -- - ~ j (Fj A,_) C: C,. ~ ,-t,t1(A~ ·Ll.i) F(Ar•A,- J Ji =, ........, for all "JI ;;,, -Yo J oO •o that ..Z: j (sA~) = J. (~ E). ,1,.:, F(~} and G-(;J;,) are boundedly integrable over £ then c::( F(J;J and F(J:.) f- (;.{;I:) are integrable over E and J ( al F ) = al j ( F) :, . J. { 1= -t- G-) -= j ( F) t- J (G-J . Theorem 2.13 Proof: about If The statements about oC F(~) F ;- c. are evident. To prove those , let f n; J and f rrt 1 be sequences of partitions of E into sets of :tJ , such that J-rr;Ct-) ~ J( F) .Jrr~ ( CJ ~ j CG-) C. as >- and ). respectively ➔ oo V-rtc. U • Given U , choose V so that By theorem 2.?, J,,i . lT~ ( F J; »',=:· rra . ( J c: J; F ( F) G-) C: cfut ( b-) But from the properties of convex hulls, Ci,. ( tl- + B) - CrA + ur B 2,#4 55 so that C Consequently for large enough Y~ A so that r -t- Theorem 2.14 G we have is integrable and If Ll is of finite measure, that is if 4 E ~ 1 , then we have the following law of the mean for L-integrable functions: Proof: (?q. F(A). 5. Riemann integrals and continuous functions Riemann integrals of functions on a closed* real interval to a Neumann linear topological space** * The interval ot ~ ','..(. (1 ~ !:-1'"< (I , < < ,r ~ ~ . [ «- .> (3)..) ( oc.., 19] ~ , are considered in Michal and will be denoted by ( « , (J) , and the intervals c< ~ 1'" ~ (J will be denoted respe ctively by (o.., ~J . ** Such a space is slightly less general than T. See Chap. 1, #3 and #?, Ex. 20. My results on Riemann integrals were obtained independently from those of Michal and Paxson. 2, #5 56 Paxson (1.) and (2.) and are studied in detail in Paxson (1.). However in view of our later applications, and in order to correlate the "Lebesgue" integral of the preceding sections with the Riemann integral, a brief discussion of the latter will be given in this section. If F'(~) is a function (possibly many valued) on [.«.,~] to 'T , we can obtain a definition for the Riemann integral of F(1') over to be the usual subdivisions i!',P] from the general definition of #3 by requiri ng the partitions~of Ti E:c~P] into a finite number of intervals. More specifically, for oc =1'o < 1', < • •• <1'.,.=f:l let ~~ denote the collection of sub-intervals Ll 1 = ('i'o/i',], A :a.= ('i',, 1'1].., - - - J A.,,. '=' (1',,_,/i',,J. Put ~ ( ~~) -= 0~ wi"l,~.,, x_i(Ll L{4..,.,) = 'i'c. -1', _,, and define the R1" emann ) integral as follows : Definition 2 . 3 A function f(1') ['-'~t9J to 'r will be said to be on Riemann integr able to 13 j(F):::. 5 F(fo'}dc; U and each ..-J > 0 , there is a sub- division ~.,, such that oC if, corresponding to each $ (szJ.,.) < 1f and Since we are always supposing that the space Tis locally convex, and since any linear topological space is regular, it i s clear that the content of the definition is unchanged if we replace by • As Z : j (A,.) t-(A") . "'' in the case of the L -integral of #3, it is convenient to define the (Riemann) integral range F(1') J,,s~ (F) of relative to ~, by & ~ ~ (4~) 1=-(4L) • I.,==-, 2, #5 5? Thus definition 2.3 could have been phrased in terms of integral ranges. (Compare with definition 2.2 and theorem 2.9). If we interpret the M -space £ fi.tJ eJ of #1 to be the real interval where Jb and J£1, a.re identical, and ea.ch consists of the set of all measurable sub-sets of \?" .,(S'J then definition 2. 2 furnishes us with the definition of a Lebesg ue integral of a function Ff-r} Theorem 2.15 ~AJJ g F(rj .QD. L,q-,(9] 1Q on [« J (3] T is Riemann integrable ™ then il ll Lebef\9110 integrable .QE.r ~ the two integrals [Ot!.1/3] are equal. Proof: The theorem is clear from the uniqueness theorem 2.8 when we notice that (1) every sub-division xJ,, is a partition 7f of E,r,(3] into measurable sets, (2) every "Riemann" integral range is a "Lebesq_ue" integral range, (3) by ta.king a sequence of , 7f s ➔ 0 in definition 2.3, we obtain a. sequence of integral ranges satisfying the conditions of definition 2.2. In the following theorems on Riemann integrals and continuous functions we shall, for the most part, merely outline the proofs, for the reason given at the beginning of this section. Theorem 2.lit Every ( singl e valued) continuous function on [oe,, (3] to is uniformly continuous. Proof: As in the case of real functions, use Heine-Borel theorem. Theorem 2.1'7 (Cauchy condition) W [«>~] F(-r) ..QD- 1..0 ! necessary and sufficient condition be Riemann integrab~ ~ ~ t corresponding i,Q every U ~ i s ~ sub-division .,$'11: f A.,j o+ [ ..Jf1J 4 ~ that T 2,#5 58 ~(~,,) < 't( ~ 2.• j(4a-) F(-6,,) - ~ j(A,,)F(A,) c:: U. '::: I Proof: See proof of theorem 2.10, and corollary. Theorem 2.18 If +(-r) is ~ ( single valued) continuous functi on on to 1' , then f(1') [Q(..,f9] Proo f : t.:S/ ~ Riemann integrable. Use theorems 2.16 , 2.17 and the postulate of local convexity . It is convenient to define J. ,.F(r:)d(;' ¥ ( i) "\ [.oc. J p] . ™ anv sub-interval exists and Th9 integral Q.f f(1') f'F(<:)cfc;- = s-ff(,;)d<. + J,rF(•").tc: fo..-','(,f<l"',P] ¥ (iii) s:F('-') cl{; In the law of the mean (iii), { F(1'J E p ~ f (J.. F(c)+ tt,G{c})dc §xists.) = >19 « the set G-<,,.J r(1') , Then '!!.§ have 'K' (ii) i,.F(,;-)d"' in case Let ~ Riemann integral .Qi' the functions Theorem ~.19 ~ .QW' as - J~-=('")dc; -t"f- f G-(c)dr;- J~ -< ~ - a.) Ctr r{[°'.,/3]} f ( [4'..,f9J) , i.e. denotes the range of JJ 'f' f- L~.J f9J . The proofs of (i), (ii), (iii) are similar to those of theorems 2.11, 2.13, and 2.14, respectively. the proofs are greatly simplified, since all series Here ,Z. ,l (A&.) F(A1,) are finite, and hence automatically unconditionally convergent. Conse- quently, it is not necessarily to bring in the notion of "boundedly inte grable " f'unctions. Definition 2 . 4 A function f(-1') on the open interval (.:it..,p) to 1' 2, #5 59 , will be said to possess~ derivative U there is a if for every t(1'-1'o) 1' - S >0 f at the point ~ (: (°',<9) (1'o) such that l'f'-1'0 I< S f('ro) 1"o Since T is a Hausdorff space, a function -f(~J derivative at 1'0 then ~ each point gt f(tf') :t,.Q 'T1 and [°'..,~] i_s continuous .QD. 'J (1') = r-f(~) d" J <J '(6r) = can have at most one • lf f(1') Theorem 2.20 implies 9 (1'} has 1.h.e deriyatiye (~ .J ~) • Proof : As for real functions, using theorem 2.19. Under the conditions of the theorem ~ (1') is continuous EE_ ~.,{:J]. Corollary: Theorem 2.21 If ~ function f-(-r) each point of (oc 1 ~) Since ~, p] .!£ 'T' has ~ derivative at which vanishes identically on («.J ra) then f (1') C'f . is a constant element of Proof: ~ +' (1') = 0 for all 1' f (ot,, f3) , there exists for each U (we may suppose U convex) and each '1" E («:., (3), a positive number such that f(ti) - f ('r) ( 2.5 ) (; - Hence since ( 2.S l a "'T" f f (-r)- -f ( 1) u) "'i'- E u U is convex, we have +(<;) (; - t(~) j f '1"-"i' ~ - for 'i Let ( ~> -., ~ u+ 'i' - g u '"- fj S('l') < 9 < 1' ( ~ < 'i' + S (1') . ~l be any closed sub-interval of ( «,, <9) . We wish to show that 2, //5 60 . By employing the Heine -Borel theorem for linear point sets we may assert the existence of a finite covering of 4c.= (i'"'-i", 1',.+a,.J by open intervals (1.) S" < s; (1'..,) [r.., "f 1 with the following properties: (so that we can apply inequality (2 .5)) and Sc, < Y,. S ('() (3.) Ac, (4.) has 'lf' as its mid-point. The intersection of the left open half of Ae, and the ri ght open half of .61. -, ( 5.) contains a t least one point Some .6 r. , say .A,, , contains If '1(_ is not the mid-point of 4,, '°"' 1f . , put -for •~ C.$ "JI select 6"1,, in accordance with (4. )A, and we may clearly suppose <;,_, < c;~ ., L =.1, • ·-, ,,.,.j . Then we have by inequalities (2.S") and C: u (2 .5 )a that f(c;-'") - -f(c;-.__,) If "( is the mid~point of for <..-=1.,· · ·.,-Yt-( . A,., , we can stop t, at ~ instead of -,,+ I In any case we have But since U is an arbitrary nei ghborhood of 8 and t1 - le/;- 0 we , q.e.d . Theorem 2.22 If +(-r) e xists and is continuous ~ (ol) fl) , then, ~ ~-> ?J E («,_,fJ) -f-(1) - 1 VJ_e ~ f (.r) = . 2, #5 61 Proof: See Theorems 2.20, 2.21. Theorem 2.23 [ce.,pJ If ff.,, (-r) J ~.!:.sequence of continuous functions on to any linear topological space, converging uniformly_£!! to .!:_ function Proof: f (1') As for real functions., , then f(-r) is continuous .£!! ~., ~]. In this theorem the postulates of local conveH ity and sequential completeness are not used. Theorem 2.24 If f f,,(f')} is~ sequence of continuous functions on to 'r converging uniformly _£!:. this interval to -f (1') then I ,·WL ">' ➔ 00 Proof: Use theorem 2.23, the law of the mean, and the local convexity and regularity of 'T" . If f(-r) Theorem 2.25 is a continuous function on topological space then its range Proof: Let f: f,t.., - - I) f ([r:ic J~J) f f z, --1., i.e. to any linear is compact in itself. be an infinite sequence of (distinct) values of the function -f-(1") , and A,..> A~, ---f; 0 [°',fl] the original sets corresponding A,.,= f 1' J, "f' f [a..,f!3] , f(-r)=fc. , By the axiom of Zermelo (used here for only a countable number of sets) we may choose 'f,, e Ac. , obtaining an infin i te sequence such that -f-('i e,) =-f ._ point f {1'o) and since -f (1') of distinct ) By the Weierstrass- Bolzano theorem there is a limit point 1'o of the set Evidently 1'11 f ["'~Pl C,;-., 1 ~ { 1"'" l is continuous, the corresponding of T is a limit point of the set l fc. j 2, # 6 62 Corollary The range of ~ continuous function on [9' ,t9] t opologi cal space is a bounded set. Proof: Cf. theorem 1.6. to a linear 3, #1 63 Chapter 3 Existence Theorems for Functional Equations in Linear Topological Spaces 1. K-systems Let there be ordered to each element X of a linear topological* K(x) space T a set K 1. Each subject to the following conditions . K(x) K(x); K2. X<: is convex. 8E t((x) K 3. K(Cl(X) = « K(x) K 4. °1 f K (x) K 5. Each -If x 1,8., « >1 then «X E K(x). for implies « ~ o. K(1) c K(x). K(x) is bounded. Then we shall call the family fr sets K (x) a K-system for the space. If instead of K 5 the condition K 5a, Given any neighborhood U of e there exists ~ neighborhood of 8 such that Z K(x) c U Jt fV is satisfied, we shall call f K (X)} a continuous K -system. It follows readily from K 3 and the continuity of scalar multiplcation that K 5a implies K 5, so that a continuous K -system is a K -system. each of the sets K(.x) When is closed, we shall call the family a closed K -system. In the case of a linear normed space, the spheres II j ll ~ II XII * As in chapter 2, we shall denote both the basis and the space by the same letter when there is no chance for confusion. It will be explicitly stated when postulates (II.) and (III.) are required. 3, #1 64 form a close d , continuous l'( -system for the space. In any linear topo- logical space the closed line segments joining the origin to the point form a closed set K-system li 0 which is "minimal" in the sense that each K(X) of an arbitrary K-system contains the corresponding member 1i0 • Ko()() of the system Ko(xJ being merely the line segment o<X , 0~«-~1 .) For locally convex linear topological spaces }< -systems have a special si gnificance. In particular, the minimal \(-system 1( 0 is contin- uous for such spaces. We now state a fundamental exi stence theorem for K -systems in locally convex spaces. In ~ locally convex linear topological space T Theorem 3.1 always exist complete neighborhood systems U : (UJ ~ W, ---) there of the origin with the properties Uf U (i) If (ii) If lj G U then U is convex then ex Uc U . For any such neighborhood system U define -#- • K(x) = TT (U) Xf: U, UfTL, K(x) Then the sets ------- space T , and~ have Proof: form~ closed continuous K -system for the I< (x) = • TT (U). JCtU,UEU. IfW- is any complete neighborhood system of the origin in the locally convex space T , there is ordered to each W € 7,()- set V c W with 6 f V , by the definition of local convexity. the family { v1 vf w.> Wl '1JJ.- forms a complete neighborhood sys tern of the a convex open Then ~ ori gin, with each V -1:- i.e.. K(x) _ convex. Now add to f v J j,,f:ersection. of ci// U eU the sets U = «- V -fo...- w'1ick. ~E U . 3, #1 65 where a. ranges over the positive real nwnbers, and V ranges over f V} _ U By t heorem 1.2 of #1, Chapter 1, each U is open, and clearly each The system U : [UJ, ~--<ti.,,.,£ is convex. satisfied (i) and (ii), and vc:Wew the first statement of the theorem is true. To prove the second statement note that, due to the continuity of scalar multiplication, for each is an with « X € U ol;,, a Hence K(x) = TT (U} ,rf(Jf-«, X f- 'T' U ~-,/ u., and each there , that is xt1/QfU:::U'E7,l, is not empty and in f act contains X~ O. We now verify that properties K 1 - K 5a hold for the set I< (x) • For convenience we denote by uce,x) Ad It 1: K (x) Uf any u with X ( U. is convex, since the intersection of any family of convex sets is convex. Ad K 2 l Obviously x~ 8 f K (X) such that • XE U • Then put ¥ Choose '1{ so as to sat i sfy Y«. ~ "'- <f'..:::. 1 "'£hen x f V since ~ 'x € o< ~j_ Take J<-:1- 0 (/ , and put Ii. X € and therefore (r«)X E U y',l ;, t' K(x) = TT f U(e,,oJ c: V;, oc. d = 5Uf f!,J PJ< E U. o , but ' an V V = 1/<l' U. since But since x E K (xJ. Ad K 3: ex.. K (X) = «. TT ~U(9) ><)) =TT £« U(e.,xi'!J = TT (U(e.,~x)) Ad K 4: 1f t: If K(X) Hence every , then (/{IJJx) ~ f- U{IJ.,x) for all (/(9✓ ,U) , and is a U {e, J&) ~ u,, 7T t (/ (6., ,t) J C TT f U(8., X) ] . Ad K 5a. Given any neighborhood W of the origin, there exists a Ut,U 3, ;l/1 66 such that Tha t i s Uc W . Then if x € U , TT[ U(9_,x)J c Uc:: W. ..2::!u KO<)-~ TTf U(e.,xJJ c:: W, q.e.d. JC£(,/ J(t: In order to show that K<ic) is closed we first establish the following lemmas which are of interest in themselves and should perhaps be included in Chapter 1. If 'T' !._s ,P. line ar topological space and Q- is _a convex ~ Lemma 3.1 ~t of 'T' ~ contains the origin, ..thfil! a G c G- - for all ex: , 0~ oL <. 1 Proof: o < o<. <. 1 If o( Let x , then by hypothesis G- -r (1 - o() G- = G- be any point of {« G) =ex G . Since °' G and (1-«JG- open sets lfontaining e , there is a nei ghborhood LJ¥ of J( are such that , by the cont inuity of subtraction. c.dOtt Since X is a4 ~ point ·. of « G there exists a point ~ E ( ot G ) • U x • Hence we h8.l7e )(. == 1' + 1' -t That is (;1(. Lenuna 3. 2 taining C: d(;. --r Ux- u)C C «: G- -t ( I - ...c) G- = (;-. G c G- _If U (e, x) is ~ convex neighborhood of the origin ~ - x , and o<. is ~ positive number less than unity , there ~ ~ number (3 , °' ;f. t3 .c. 1 Proof: Either the lemma holds, or there is a number t", o < lf"<.1. so that 1'f ~ 1 . such Then X is in the frontier of the op en set 1'( U(e., X) . • 3, #1 67 by continuity there is a ~ > 0 For if VJr is any neighborhood of X such that ). X E Vr satisfying for all ). € ¾ U(6 , x) )\. "> 1 implies ). > 1 implies J< E 'ti/>- U(eJx) each v,Y J< , or I >- - J. l 4 S . ).)( E l( U(BJX) But , while or ')..x E ~ U(e.)x) . Hence contains points of ..,, U(e.,x) and also of its complement, and X must lie on the frontier of 11 U (8.>X) where U(e., .x) whence '1J < 1 is open, X . . But since )( € U(0.)x) U (9 > x) is not in the frontier of By the definition of 1f , J< € l9 U(9-J X) for , and the lemma 3.2 is proven. K (x) =IT (0) To demonstrate that K(x) and therefore that is x~v,Ufll, TI (0) c:::: l<. ( X) . Suppose that ' for all Uf -u such that J( ~ u . Let u' be any closed we first prove that Jtf:l7,</~U '1" €- u member of U such that containing X • _, u' - By lemma 3.2 there exists ""' «. By hypothesis ( i), «. U' X € - is a member of -u , call it U ",.ci_x) \ o-, . Using lemma 3.1, we obtain 1' f (/''(6JJ,9 , so that '7 €- (/ TT ( u) c::::: TT (U) = K ex). Now by hypothesis, that To demonstrate that ( _ X E U . for TT (U) ~fUE-1,{., l<(X) C T I (U) suppose that JCf-u,,vt--tL , and let U be any member of -U, such that We must prove that 1f € U , hence X E ¼x LJ E -U o < « < .1 , and it follows IC f U, (H 1,/, ;c£:.v.)11eu. ~~ l<.(x) = / • . By lemma 3.1 Since 1/ E K (x), 3,#1 68 w:e also have 1" f ~ U or "''t f U for all «., 0 <. r,t ~1. Since scalar multiplication is continuous, i.e. ~ E- U f r . is a limit point of lj , Combining our results we have that TT (U) TJ. I< (X) = Xf'fiJ 1/E Also, since the intersection of a family of closed sets is closed, t<(X) is closed. In view of the later applications to functional equations it will be of interest to examine some examples of K -systems in a few of the important instances of non-normable linear topolog ical spaces (cf. #7, chap. 1). (A) In the space 1> of infinite dime nsional ve ctors the sets K (x): l..tj' f , 1-f, J ~ I Xc.,I, t..= I,':,---, form a continuous, closed 'K-system. In the space ~ of all real functions the sets (B) * { j l., (' (f)f~ l+(fJ/ for - oo <. \ l< (f J: <+ oo~ form a continuous, closed K -system. In the weakly topologi zea Hilbert space ~.... , the sets ( C) * K (-f): f 3}.J I (j,ct>)\~ I (-f, <{))/ for all cf>, form a continuous, closed K -system. (D) In a weakly topologi zea. linear normed space i 1 J:, f -f. ( j) / ~ ( f-(~JI continuous, closed K -system. locally convex, and the set e w , the sets l< (x) : for all 1 inear functiona ls -f., form a In each case, the space considered is \<(x) is the intersection of all those neigh- borhoods containing x whli:~h belong to a complete neighborhood system of the origin having properties (i), (ii) of theorem K(x) 1. For example in (A) is the intersection of all the nei ghborhoods U( 9 ~ ~.;~-., s,..): * The spaces '?-- and f:./ 1111 are non-metrizable as well as non-normable. Cf. # 7 of Chap. 1. 3, # 2 69 as -;,, varies over the positive integers, and . b, varies over the real numbers greater than From the general existence theorem 3.1 it follows that the families of sets K(x)given in (A) - (D) are continuous closed 2. 1' = f( 1) . The equation Theorem 3.2 .f-(1) be ~ continuous function~ V- to V Let ~ !. sequentially l.K(-t)J K -systems. where Y complete sub-:-s.et of !. linear topological space 'r . be any \( -system with respect to which the function Let fCtJ satisfies the "Lipschitz" condition for all pairs j'J z € Y. Then the transformation z == .f (1) has~ unique fixed point. Proof: We employ successive approximations. "1'~-t-1 each f ( f c,)., a. ::: 0 -= is well defined. - Having chosen ,t'o f-V , put .JI J 2'., - • - Clearly Using K 3 and K 4 we get -f(~,)- f(to) c: f- K(r,-10) - f<(f'Ct,-"toJ) f Ct:1-) - -f ("t•) c: ~ 1< ( 't-- - 1f) . c f-' K ( µ (,r, - ~a)) = fl 'A. K (-i', - ,to), and by induction we obtain Consider the series # 11 -t ,d,, rfi.,~ + -· • for <. = 0 ) I .) 3.,.I - • - . where .,d,o = tf'o , ,1,1,;, .,., = i f,-11 - 11J A Cauchy remainder /i.-,,.> j =- fo-,,+, -t- - • • ..,_ fo,,t-$' 3,//2 70 satisfies A-,,,j C µ"" K ("f, -fo) T~~+I l<.(f,--t,J + --- + F""+~-• K(f,-f,,) - ( /'-.,, + • - - I( ( ~ • -.-<fo) since K ( 1"' -fo) + fl.,, rs- - , ) is convex by K 1. K (.-cy, -7 0) But since by K 5 the set is bounded there is a ~ :>o corresponding to any neighborhood U of the origin such that o..::. f-, < S implies (3 I<(_.,, -~o) c LJ . By hypothesis o-<. µ.~ .f_ , so that QC) _.z::! µ" converges, and so f or all sufficiently large -,, , L =-1 ./L~ 'I C p I< C,t, -"'to) C U for all complete, the series .d-() + ,,d/, -t- - - - 5' . Since Y is supposed and therefore the sequence 1'' of Y . But "'f' = -f ( ,r,. -•) and f is continuous, so that f(-"t,.) -, 1f' . Hence 1' = f(f') q.e . d. f o j',,, - - - converge to an element > To demonstrate the uniqueness of the fixed point, let z solutions of z. - w Now fl where 4 1 Cl{ > 1. Remark: 1' = f(t) with z c Y, w € 'f and w be two . Then = -f(:£) --f(w) c- f'-' l<(z.-w). by hypothesis. . Hence Putting P<... = 1/ f', :z. - y,/ = The property ~ X E e K ( X) we have o<. ( x.- w) c t< (:z: -w) by K 2. for « "7 .1. I X :/= 0 is not used in establishing the existence of a solution, but only in the uniqueness proof. 3. An existence theorem for~ differential equation Theorem 3.3. Let T be~ locally convex, sequentiall y complete linear topological space and let cl, be ~ complete neighborhood system of the origin with the properties 3,#3 71 U EU (i) Each (ii) u € -u is convex implies oc: UE U - - Let I be a real open interval, G- an open set of 'r --- - - - - ----- --- -- continuous on LG- ~ T If -f(-r_, f) is and satisfies the "Lipschitz" condition ( I'- a constant > o ) and "'f, z e ~ may G- and a. E G assert that (1) The differential system 4. d}/r ( 3 • 2) = a.. = Tr ,~ \.. , ,t,J,.) 4 , ,~{«J I has at most one continuous solution wi t h values in G . --- - - -- ----- ---( 2) S >o There exists a (3.2) has a continuous solution to Existence proof: such that the differential system 1 =1f(-r) on the closed interval G-. We seek for a continuous solution of the integral equation (3.2)a where the integral is of Riemann type. Since f(~f) is a continuous is a continuous function of 1' for function,_ of 1', "'f, a continuous 1 (1") (Cf. Alexandroff and Hopf (1.), p. 53) by the corollary to theroem 2.20 the functional transformation f fC~,rc<;J) de 1' (3.3) z (/fJ - a. +- -< Hence 3,#3 72 takes a continuous function 1'(1') on on I to G ;.. t-o a. J1 continuous function z(1') I . aE G Since where G- is open, and since the space T is regular and locally convex there is a convex neighborhood a.. -t- V C G . Choose f3 E- I V of e such that so as to satisfy simultaneously the inequalities and keep (3 fixed. Now if ,t<c') - a. f V for Ci f ~,pJ we have, on using the law of the mean for Riemann integrals (Theorem 2.19, (iii)), that z. (1') - a. Therefore, since V , and hence V, is convex, we have by inequalities (3.4) that for C That is the (-r - o<) .!. ((3 - oc.) :z. V ;- ..L V .-:l. functional transformation (3.3) carries the set Y of contin- uous functions 't ((;:) satisfying ,r(c-) E- a+- V a part of itself. ( 3 .3 ), carrying for '- E [«,, Pl into It remains to show that the functional transformation Y into a part of itself- has a "fixed point", i.e. that the integral equation (3.2)a has a solution. We shall prove this by the method of successive approximations, after a few necessary preliminaries. Put K(J<) = TTCUT.l(U) )(E l<(Jt) Then by theorem 3.1 the sets~forrn 3,#3 73 - a l(-system for the space T , and we have 1<(x) Hence for any U E U we have Let us denote by K(M) the set K (x) K(U)= U fc. (1') XE U , where M is any subset C: KCO) ' so that (. 1-J - - - U '= U . s:f ,tJ f 1',..~, (1') = 0.. + for u inductively by 'to ( -r) = a.. (3.5) for each ..,Z K(x) ,K~I,,( , we may also write for each Define X( U _, 1./f 7/,, Then, from what was ~ust shown, K(U)c U of our space 'l' . But since x f I< (x )c:. LJ =TT (U) ('"J 6 )) d..t:: .:=. ,/ OJ 11 1'f («, ~]. Clearly ,r,.(1') ~ Y for each <- , so that for each 'f'f (-=, ,9] have 'fc..c,r) f a, + V we In order to prove that the sequence "fo <-r}., -At,tr),f&(<t),··- is uniformly convergent for 1" E [ot J (3] arbitrary nei ghborhood of the origin . , let W be an Since U. is a complete neighborhood system of 0 , and since "r is a regular Hausdorff space, there exists a with lj€ W to 'T' a bounded set. . Clearly "f,(1')- j 0 (1') is a continuous function , and by the corollary to theorem 2. 25, its range is That is there exists a positive constant 1 such that for all 1' f [°'-., (3] This statement is evidently true for -v= o is true for -,, .::c " - 1. . By an induction we , so let us suppose that it On using the law of the mean for integrals and introducing the Lipschitz condition (3.1 ) we find that 3, #3 ?4 -1,~, ('r) - 1' (~) - J(f. ( '-, "r,c,J) 'i' .f-( ~, 7,-, (c)}) cl~ ,< where, of course, K (x) = fr ( U). J(~(/{-U, The induction hypothesis states that are both homogeneoi:.1s wi th respect to positive scalar multipliers we obtain t'- &- K ( ~ µ'-'<P-«J'-, U) c. r(p -or) Gr t<(U} -1'at (1') - ~,,(1') E ((!:> - ol.} (.µI, = ¥(13 - o<.}' I'-' &- u .:= Y( -.1.).,f-' u ~ The last equality is true because the closure of the convex set U is convex. The induction step is now established, and it follows that for Since '11 = o,, I->~., --- "'° ((> -ot.)"p," f3 satisfies the inequality (3.4), (c), the series ,Z , ts convergent. ... Let K be an integer so large that .,,, ~ tc implies 3, #3 75 Then we have for any integer v >t< any positive integer f and for all 1' f (Cit', P] that j'•I '1,, .. ,(1') - -1,,(1') = -4.t (1f-i1-H.+1 -,t-,,+c.) r .Z <~ -«:J'>'+' fL.,, ♦ I. u -9 ·• E ,co LJ C: C The last t wo steps are justified since W. U is convex and contains fJ , But since W was an arbitrarily chosen neighborhood of the origin, "to(1').>1•(1')., --- is a fundament al sequence for each the space is complete, and the choice of t< it follows that the "'t.,,(1') 'i' f (.«., fJ . Since was independent of 1' f- («, (3] converge uniformly to a function w(1') which is therefore (cf. theorem 2.23) continuous on [«)p] On tak:ing limits · in both sides of (3.5), recognizing that ~(-r.., 1,,(-t)} -+ 4(1', we.,,) uniformly, we see from theorem 2 . 24 that 'I' W(1') = a.. ;(c:, W'(•J)d~ . sf e(, Differentiating both sides, making use of theorem 2.20, we have .f (1'., W(1')) where vl(o(.) A f or each = a,_ similar proof shows that there exists a (3' f I and a continuous function z (-r) ' fJ <.OC: ' on "'i' f [p:«-J with for each 1' f (f:«j z (oc.) .::: a.,. , , where 76 The conclusion ( 2. ) of :the theorem now follows on putting a= 111in (1(3-•l,1(1'-o(.I) Uniqueness Proof: on («-S.,ot] [°'-,°'+~]. on and 'f('l')=w(1') = z..(-r) -,c1") To establish conclusion (1.) of the theorem let "'(,CT) and 1'., (1'} be two continuous -solutions of the differential system (3.2} on I to G- . Either 1f •(-r) = 1':1-- (-r) at some point ~ € I . We may suppose -C~ > 0(., similar proof applies. Put (5 = s "r c:;, so that « ~ (3 < 'r. Then there is a 2S'" , / ' ' (1"o) - +(1'., ,fc. (-r))) 1-· (1'J ,= "f..(1') Ol'C 1 / O < 11"- 13 < C°'. ,: ] ff, , at some point lfo E- [ p., tJ --t"-a. (f'oJ '# e 1"' (1') hypothesis the func t ions for if ~ "- «. , a (. = ,., a, and ~~(1') are continuous *. such that . By are continuous, so that Hence by theorem 2. 22 ., ,r,.<-r) satisfy the equation a., + f 7-f ( c~ -f(c-J) d c -r ~ I . .t Thus, by our select ion of (J, there is a nei ghborhood U of the complete nei ghborhoo~ system U of the origin such that Now the function -'(, (1) -1',. C,,r} is continuous on the closed interval * Cf. Alexandroff and Hopf (1.), Satz III, p. 53 . 3, #3 77 [f~2f] , so its range for this interval is bounded (cor. to t hm.. 2.25) and there is a constant 'tf > 0 such that -for o. II -r E [ (3., rJ. Using the law of the mean and the Lipschitz condition, exactly as in the existence proof, we obtain for 1',l-rJ - --;,.<-rJ 1' E ( '3, 1·] , = fr( f(~.,"f,{ii'J) - f(<:.,1'._c,1))c:1(;' E (~-p) " (3.6) Co- Z µ. K ("f,Cc) -"t._(cJ) <:f [ (J--~J Hence, since U is convex, and K (D} = U for all -r E (13,A"] . Substituting this result in the right member yields "1• (1') - -f>-('T') E· ( Y-~)a. r,2---, U for 1' f lf->, l' l, and by induction, it is easy to establish that -t,(1'1 ---t.(-r}E (Y-p}'-f-'1 U for all i'f- Lat, f3] 0 4 and any given positive integer 1.. • (!°-(3 ) f- 41 (er- - (3) .. ft 1/, , -1'• ("'°) - , so that for sufficie ntly larse L , i. ~ But . Thus for sufficiently large L , 1' (1'o) € ( ¥"- ~)" ~1,.,, lj 2 contrary to the hypothesis that on C u~ "'{• (~) i= 1,. ('i',) . Hence I , and the uniqueness proof is complete. 3,#4 78 4. An application of the general theorems By specializing the abstract space 'r we can obtain existence theorems for various func t ional equations f:rom the general theorems 3.2 and 3.3. We shall consider below an existence tti.eorem for an infinite system of dif'f'erantial equations, obtained by interpreting T space* V- of infinite dimensional vectors 1< = as the cx,,A:,._,- · -) . Since 1f is Il8trizable, a function j(1'.. X) , where 1' is a real number, X a vector in 7J- and the values of the function are also in V , is continuous** if and only if real sequence 1".,.. ➔ ,P g ( 1'-,,.,.X'>'}_,.. ')('i',x) and any sequence of vectors In terms of the components of the vectors -' and x continuity reduces to: / {1'., ;<) X,, ~ for any x . , this criterion for is continuous if and only if, for each In the space V--, consider the complete neighborhood system 7i., of the origin consisting of all sets 'l . t 4 ~ I <- a,. ., U ( s,., -. -., ~ ..., ) : ts- "1o ~ O J - , A = I, '-., ···.., -Y, obtained by letting the ac.s run over the positive reals, the 71'.s over the positive integers. each U {S,, ·--.,, ~-,.) As we have already remarked in #1, is convex, and we also have = U(oe~,., - --.>«5,.) t U for « > 0 d. U(S,., • ··., ~ J.J) , so that conditions (1) and (ii) of theorem 3.1 are satisfied by the neighborhood system -U . the set K(x) = TT 1,/.( U) Evidently is identical with the set ICfUf *Cf. example 1°, Ch. 1, #7 ** This is the "Heine condition" for continuity in metric spaces. Alexandroff and Hopf (1.), p. 58. See 79 Theorem 3.4 Let G> be ~ sequence of open sets of real numbers, all but a finite number of the r;: s consisting of the real axis (- .., .1 +- 00 ) and let I be a real open interval. Let fc. ('i'., JC,., x • .,---) be ~ infinite set of real functions of ~ infinite nwnber of real variables, and for each l , suppose that -f-t.(1"'~;c,,X,_,---) defined for 'i'C- r ' /(A ~ G-.,. is and is continuous in the sense that Tt'('f'IC ., ,.., x.,..., - - - ) . for all 1' f- I ---- .> J<.>. f f; ,. Finally, ~ suppose that for each c. , -fc. . satisfies the Lipschitz condition \ -f-c.('1.,"f,, ,t~. · · · ) 1 for "tc I, 1'1o., 2>. c- G-~ I, • ~ ~L(-r,z,.:;i;,._.- ··) I ~ where ~ 1 . -.z. .. l fL 1 ~!!positive eonstant independent of Then (1.) for~ chosen ~ f I the differential system with has at most one set of continuous solutions "t"c. (1') E G- c. ( 2.) for 'i' f there exists a I . S >O such that this differential system has ~ set of continuous solutions 1"'- = A(c. (T) for {1 -o(... I ~ ~., a. where "'f'< er) E G-c.. Proof: Immediate from Theorem 3.3, the above considerations, and the for 11' - al. I -<. 3, //5 80 fact that iJ is sequentially complete (cf. discussion of ex. 1°, ch. 1, #?). 5. Completely continuous linear transformations ~ t 1' = {-('-) be a linear transformation of a linear topolog7 cal space rr ,,into a sub-set L of rr the origin who~e transform -f-( U) that f -,f . If there is a neighborhood (! of is a compact set, then we shall say is a completely continuous linear transformation (function). It is clear that a completely continuous linear transformation .f takes bounded sets into corapact sets, for if 5 0( is bounded, then for some real and ..f(s) cf («U) - «+(u), whence -f (SJ is compact. Thus, when 'f is a linear normed space, the above definition of complete continuity reduces to the usual one*. We have proved in theorem 1.9, chap. 1, #6 that every locally compact linear topological space is finite dimensional. This suggests the possibility of extending the Riesz theory of completely continuous linear functional transformations to linear topological spaces, with the above definition of complete continuity for such spaces. I shall show below that several of the simpler theorems of Riesz concerning the equation is co ... p let"~ 1-, "'f = X --f(lc) <0KT«°,u1ous where f (.('J ~ are inllrediately extensible. The following theorem illustrates the power of topological methods and will often permit simpl,ificatfon of Riesz's proof~. What is more important for us, it will make it possible to eliminate the role of the norm. 3, # 5 81 Theorem 3.5: If .,,,=.f(x) is ~ completely continuous linear transforma- Cf' into !. subset L of -r , tion carrying !. linear topological s:race then L is a finite dimensional linear manifold. Proof: L is obviously a linear manifold, for if "'f• f- LJ -<f~ f-L J where 1", ~ +(,c,) , "/1.. = ..f.(IC.,-j , then o( ,;, t- -'-q>- = + ( dK, +,,! x.a) E- L . We wish to find a topology for L which will make L a locally compact linear topological space. Let U be any neighborhood of the origin. U-,,. :::E ~ - t .f-(U) . ~ f: L For any put U1" as fundruoontal sets, L Then with the sets forms a linear topological space. Verification of the postulates =========- = = {I a) Obvious, since {I b) Given U ,I/ and Cu (} € U U_,.,,= ~ +- f{<I) ·U' +(u,,) , choose Then clearly f (u I) . C / U"' = t (-j-t--f({I)) • ( 1- -t-.f.(U')} . 1-_ _, / ...... , That is u,; c: Uy/ u-t. -t- f ( (/ 1/ } / c Hence (I c) If .z: €- U1-= 7 where x. € u -t--f(U) ·; 4wce then -z.-= "'t t- f(;c,), u so that ,c, t- u' I f is linear , i ~(x,) + f(U') = whence ~✓ . ~ -t- -I-( u1"' '4 i.e., <jz. c::::. f u C -f(x,-t-U') c -/ (t-i}-t- . Then, since f(U), -t(U') c -'d° -r -uu) U1' . ( I d) / f f r t - z ' ~., .% e L c::. T neighborhood V of the origin such that z - 7 there is a £ V . Since 3, i/5 82 I +(x) is continuous at a there is a u such that -f-(U)c V. e -f.(V) He~' z - ~ U,. = ~ -t- -f(U} , or, putting ....., LJ"t . € we have ',, Z (II a) '"' Giv~ U \ 'l-t-z. =.-4-+-Z -t-f(U) - ' • , choose V so that V + V c U Then, since f 'is linear, we have "· 1' T -t- ( V) ~> ~-, -t + ( V) = 1-'" + :z. + ..f. ( y -f V) Vy + That is, (II b) 1' + z. + + ( u) ad' -i-Z. Ua."o::::. al-ff r-t(U}, let the real interval Given neighborhood Vx. ' ~ ' C U I of (}- be such that (3 f- .ld. (3 (X "t" U ') f « X -t- where UJ I" and the implies 1' -= -t (;<) . Then, since -f- is linear, we obtain (3 -f (,< -t- U ') = -f ( (3 (x -t U'J) c f («'J( +- U).:. «j ...-f(U}, (3 (-f' t-f (U')) = for all That is, on putting (3 l-- Iae u; =1 - .-., + -f ( u '), Io( u1 C: u"'1- Thus L forms a linear topolo gical space £ if we take tre sets as fundamental sets. But since f(x) is supposed completely continuous, there is a neighborho;d U whose transform -f-( U} is compact. Hence I:. is locally compact ; and by theorem . \ 1.9, chapter l, #6 / the "basis" L of L is finite dimension~l. Theorem 3.6 I \ If/ ~ = f (;c) ~ ~ completely continuous linear \ trans - / a . \ fonnation of i linear topological space 'r into !:. s c,b-set L ofrr.: t han I \ the equatiq£ x - f Cx) == e has only a finite number of linearly \ / independent solutions. Proof: The set of points J< for which X - ,f (JcJ;::: 0 "forms a linear 3, #5 83 s~pace M of the finite dimensional linear space L which is the f == f(x) range"o, the transformation . ~ If -f(x) is ~ completely continuous linear function, "-, and if the equati'Q~ 1 = x - ~ (JC) has~ solution for each -f , then Theorem* .f (x) = 0 the homogeneous equ~~on }( - ,, x= a . Proof : Put ·-,,. has the unique solution ·,, = -f: (x) ·", ,,_x - .f (x) t-.,,'(x) == t ( tc.-,, ~~ { x))., t'(x) -v = ;z., 3.., - - - '", Let /V1,, be the linear manifold cons~-~tJng of all the elements ,x E 'T' for which t'.,.) (x) = e for which t(x,J = e '"·" Suppose that tJ1ere is an element JC,-:/- 8 . By hypothesis the·r~ is as elution JC = x ... of '\ , a solutionX=X3 of t:-(;c) X=J<.,, ot -t;-(;c).:::: X-v-, t<-,,J (x..,,+, ) Then That is x.,,,., € = x,._ '· \ , and a solution = t(x,..)., 7-=.l.,3.,· •-. , while t; ,.,, •• , (;~· ♦.: ,) = 0. . But obviously '/t.1-,, c. M,..,.., , for any chosen ~ , i.e.·\-!f.,,-, = ;< :fa 8 1 M1'-tt ..:. M.,, sp that M.,, is a proper subset of Nf.,,+1 ~ \ -for .,, =- I.., "2-~ - • - . \\ Using operator notation, where "e" denotes the identity operator, we \ \ have \ t',,>(x) - ( e - -f ),. x - - _:±1 Ct ) (-f)".:::: =, ,,¼ and where ( -::) =• 1,~(-,,-t.J! where 9 .,, - C,, "\ = X - 1°°" (Xj + (~ crJ(-f)•-· \\ (e - ,,,)x ,;. \ Clearly then, the range 'J -,, ('T') of the completely continuous transformation * Cf. Riesz (1. ), p. 82 and Banach (1.), p. 153 . , -,, is \ \ 84 contained in the range + mat~ '"' f (T) of the completely continuous trans for- and therefore, since M -v c:::"f (rr) M71 c . 'J,('T') , we have / Now by theorem 3.5, the linear manifold~~ finite dimensional, so that trere must be an integer fL such that /VJ~\ = Mµ,t-1 , since a finite dimensional linear space ', cannot contain an infinite sequence of linear manifolds, each a proper subset of the following one. ~.+ B Thus by supposing arrived at a contradiction, and the theorem follows. tally proved the We have inciden- ' ', '·,, ,, There is a non-nega,t ive integer :.,, such that for ~ > "J' Corollary*: every solution of the homogeneous ~qua ti on , while for ~ has ~ solution not satisfying Proof: we have t<,._J (;<.) == 8 O ,~ - fL <. ..,, t ,,,., ( xY~ , ~ equation -t; CIL+-l}(x) = 8 e. Take -,, to be the first integer such that Put t-(x) Theorem 3.8 = JC - -f (J<) also sati s- /'A.,,= M ,,.,., • , and denote & L f'- the range of the linear transformation t</A-J (x) exists ~ integer 1' such that !£E. r,- > Y,, 0 $ I'- 4 ""'~ L -Pt-I Then tmre L ~ = L.,, > is ~ proper subset of L,, . but for This integer -,, is \ t h e ~ ~ that of the preceding corollary. \, Proof: Clearly L,-c, , being tlle ral\ge of a linear transfo~mation is a / lfnear manifold, and si nee I L IL r- +I C l L rd • Al so' i f f- (/A--+«J ( rr') = j; Ct') (-t ( 'l'Jj ,\ we have L ,... + ' = L ~ , then L..r = L,... 'Th.us, if the theorem does not hold, then every L/A'.-, \ is a proper subset of L,.._ . Therefore there exists an element X~ , in for al~ ' S 7 fk . * Cf. Riesz (1.), p. 81 3,#5 85 )'. I"-- - )( where x = -t-c /C 1-'-J ,,, t; <~µ,T', J <:- L ,,.. r p.-t,,,, Since .K11,+• -t- +(l',_.,) --f; (1C14Tt) f LI"+' )( E l1-1-+1 , while Xµ. x,,._-x e L"".,.., But if L fl-+I . Hence l" since +1 Ly,+1 , we see that f (](IL - )(I'- 1-1 ) € L ~...:. LI' +-I is a proper sub-set of Lp for all fl, then we are led to an infinite sequence of linearly independent elements J c L = -f (rr ..) , with contrary to the fact that L 1 s finite dimensional (cf. theorem 3. 5) • Therefore the first conclusion of the theorem holds, and we have for /A<. Y L -y into its elf. X~ Z , but l-,, := L.,, +t Hence, for any fl.,, of the equation O f l -,,, -t: (X)=1" . That is t (x/ talces there exists a solution . The rest of the argument may be carried over word for word from Riesz's proof, see Riesz (1.) p.85. From theroem 3.7 and corrollary, and from theorem 3.8 we obtain at once the following criterion for the solution of the equation Theorem 3.9 If / .f.(x) is~ completely continuous linear function £n 'T' I to rp then ~ equation )( - f Vt) == 'o has !a_ solution for ill i ' E rr' if and only g the homogeneous equation J< ~ Proof: than J< = If J< = 6 t (JC)=~ -+ (JC)=0 has .ill' solution e is the unique solution of clearly have the case -Y = 0 of -theorem 3.8. -t (x) = 8 then we But then "t (x) takes 86 into itself, and for every -i'f has a solution. T , the equation -f;(x)=--'t The necessity was proved in theorem 3.?. 87 The Kuratowski Postulates for a Topological Space. Appendix I: Let - be an undefined set operation which orders to each sub- set M of a "space" £ , a subset M of £ , subject to the following postulates (cf. Kuratowski (1)) M -i- N .::!: M +- N (i) .J. (f) =q,) (ii) f f £ where (f) is the set consisting of a single element (iii) M c: M (iv) 0 =0 A space £ where 0 -f. is the null set. satisfying (i) - (iv) also satisfies postulates (Ia) -(Ia), and conversely, as is shown for example in Alexandroff and Hopf (1.) p. 43 and p. 59. If 'I' is a linear space satisfying (i) - (iv), and if in addition the following postulates hold for (v) M-t-N c. M -rN AM c r-r. for all sub-sets M.., N AM of 'f A of real numbers and all sub-sets M of T I where A denotes the closure of A , then is the "basis" for a linear topological space ;J , where neighborhoods (vi) for all sets U x are open sets contai. ning .K , a set U being defined as open if rp.:.. U = 'f..!- U . The pDstulates (v) and (vi) a.re simply another way of stating the continuity of the operations of vector addition and scalar multiplication, making use of the following definition of continuous functions: Let £ and £ ' be spaces satisfying the KuYatowski postulates (1) ✓ - (iv). Then a function f on E to E will be said to be continuous 88 on E if +(M) C: f(M) for any subpet :M of £ . For the proof of the equivalence of this definition of continuity to the definition in terms of neighborhoods used in Chapter I, see Alexandroff and Hopf (1.) pp. 52 - 54 (Satz IV). The postulates (i ) - (iv) are essentially those used in Kolmogoro~f (1.) and Tychonoff (1. ). 89 Appendix II: Pseudo-normed spaces. A linear space T will be said to be pseudo-normed~ f to each of T there corresponds a real number satisfying the con- fxl ditions Ix\ >,,.o ; (1.) (2.) lc,(xl \xi =o )C =a = lei( txl If l2'1 ➔ o., (3.) .:>. 11-l ➔ O then IX +1- {➔ 0. We shall show that a linear pseudo-normed space is a linear topological space in the following way: If 5 is any sub-set of 7' we denote by S the set of "contact" poi u.ts of S , where by a contact point of S we mean a point J'. such that for every E > o . (See Appemdix I ) M + N C M +-N . To prove that It is clear that M+NcM+N - +N , take X E- M . If X is not a contact point of N then there is a ~ such that no point \x-al ""'- s . Therefore for each "1\ )' $ , i.e. >< E M that 1x-;z:( < -., ){EN . Ad ( iii ) Given "'1 -;,, o exists a ~>o such that similarly there exists a l "J- - X o \ 4. i Ad (v) If ,,_ M.,, EN ... Tk ,., , . ,;r of N satisfies there is a z f M such Similarly if X f M then Postulates (ii) and (iv) are evidently satisfie d. , by property (3.) of the pseudo-norm there \x+cJ-1 M , there exists an Given X 0 E 1' E- 5 with Then we may demonstrate that the postulates (i) - (iv) of Kuratowski are verified. Ad (i) there is a E ;< f- for l )( I ., L1f l <- a . "'1 4 M such that \X - 1'o l L. 5 1' l: A-tf such that . , and Hence , where ~ €- /A , i.e. M , 1-"E- N , there exists sequences ,;WC~ f: M., with ,,yt(,.,,-)C,-> r~e11do- tt.O\o"'M. 1:> .s e v d o - """'e. -t r, c. is -f a _... .-<ov 0 , / ,+C...,, - 1 ' , ➔ o . Vo be- ( j f'l eCLr -t;" c> f" o Hence by (3.), COK--fu.socl .... ~-H.. f o 'J j c« / Sf' 4 C iS S • N ev- «HrrS 90 I Mt,, +-M,, - )( -11 ~ 0 Ad (vi) ; Follows from the fact that I CC J(. as oCr:,¥0 J = I °' X - o< Xo T«.J(o - °'oXa l ~ 0 and I a! 1 o - o(o Xo I = I~ - « o I I Xo \ ➔ 0 . l«X- o< Xol=l o< IIX-XoJ ➔ 0 From Ch. 1, #6 we now have the Theorem The necessary and sufficient condition for!. linear topologi- cal space to be pseudo-normable is that !.!_ contain !. bounded open set. It is cl ear from Tychonoff's example H,/.1. , and the results of ch. 1, #6 that a pseudo-normed space is imtrizable, but not necessarily normable. 91 References P. Alexandroff and H. Hopf (1.) Topologie, vol. I, Berlin (1935) s. Banach (1.) Operations Lineaires, Warsaw (1932) G. Birkhoff (1.) A Note on Topological Groups Compositio Math., vol. 3 (1936) pp. 427-430 (2.) Integration of Functions with Values in a Banach Space ~rans. Am. Math. Soc., vol. 38 (1935) pp. 357-378 (3.) 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The Differential in Abstract Linear Spaces with a Topology Comptes Rendus of the Varsovie Academy of Sciences (in press) D. Montgomery (1.) Continuity in Topological Groups Bull. Am. Math. Soc., vol. 42 (1936) pp. 879-882 J. v. Neumann {l.) Zur Algebra der Funktional-operationen und Theorie der normalen Operatoren Math. Annalen, vol. 102 ( 1930) pp. 370-427 93 (2.) On Complete Topologica1 ·spaces Trans. Am. Math. Soc., vol 37 (1935) pp. 1-20 E.W. Paxson (1.) California Institute Thesis (1937 ) (1.) Uber die absoluten additiven Mengenfunktionen J. Radon Berichte Ak. Wissenschaft, Wien (1913) F. Riesz (1.) Uber lineare Funk:tionalgleichungen Acta Math., vol. 41 (1918) pp. 71-98 A. 'fychonoff (1.) Ein Fixpunk:tsatz Math. Annalen, vol. 111 (1935) pp. 767-776