WIXLIB

4PART 1 : GENERAL THEORY(a) llx Y ll l l x ll IYI II for all x and y in X,(b) ll,:x x ll I ,:x I l l x ll if x EX and ,:x is a scalar,(c) l l x ii O if x # O. The word " norm " is also used to denote the function that maps xto ll x ll .Every normed space may be regarded as a metric space, in which thedistance d(x, y) between x and y is II x - y 11 . The relevant properties of d are d(x, y) oo for all x and y,(ii) d(x, y) 0 if and only if x y,(iii) d(x, y) d(y, x) for all x and y,(iv) d(x, ) d(x, y) d(y, ) for all x, y,(i)0 zzz.In any metric space, the open ball with center at x and radius r isthe setB,(x) { y : d(x, y) r} .In particular, if X is a normed space, the setsB (O), {x : l l x ll 1 }andare the open unit ball and the closed unit ball ofX, respectively.By declaring a subset of a metric space to be open if and only if it is a(possibly empty) union of open balls, a topology is obtained. (See Section1 .5 .) It is quite easy to verify that the vector space operations (addition andscalar multiplication) are continuous in this topology, if the metric isderived from a norm, as above.A Banach space is a normed space which is complete in the metricdefined by its norm ; this means that every Cauchy sequence is required toconverge.1.3 Many of the best-known function spaces are Banach spaces. Let usmention just a few types: spaces of continuous functions on compactspaces ; the familiar If-spaces that occur in integration theory; Hilbertspaces - the closest relatives of euclidean spaces ; certain spaces of differen tiable functions ; spaces of continuous linear mappings from one Banachspace into another; Banach algebras. All of these will occur later on in thetext.But there are also many important spaces that do not fit into thisframework. Here are some examples :(a)C(!l), the space of all continuous complex functions on some open setn in a euclidean space R" .

CHAPTER 1: TOPOLOGICAL VECTOR SPACES5(b) H(Q), the space of all holomorphic functions in some open set Q in thecomplex plane.(c) c; , the space of all infinitely differentiable complex functions on R"that vanish outside some fixed compact set K with nonempty interior.(d) The test function spaces used in the theory of distributions, and thedistributions themselves.These spaces carry natural topologies that cannot be induced bynorms, as we shall see later. They, as well as the normed spaces, are exam ples of topological vector spaces, a concept that pervades all of functionalanalysis .After this brief attempt at motivation, here are the detailed definitions,followed (in Section 1 .9) by a preview of some of the results of Chapter 1 .1.4 Vector spaces The letters R and ({ will always denote the field ofreal numbers and the field of complex numbers, respectively. For themoment, let I stand for either R or ({. A scalar is a member of the scalarfield 1 . A vector space over I is a set X, whose elements are called vectors,and in which two operations, addition and scalar multiplication, are defined,with the following familiar algebraic properties:(a) To every pair of vectors x and y corresponds a vector x y, in such away thatx (y z) (x y) z ;X contains a unique vector 0 (the zero vector or origin of X) such thatx 0 x for every x EX; and to each x EX corresponds a uniquevector x such that x ( x) 0.(b) To every pair (a:, x) with a: E I and x EX corresponds a vector a:x, inx y y x-and-such a way that1x x'a:({Jx) (a:{J)x,and such that the two distributive lawsa:(x y) a: x a:y,(a: {J)x a:x {Jxhold.The symbol 0 will of course also be used for the zero element of thescalar field.A real vector space is one for which I R ; a complex vector space isone for which I ({. Any statement about vector spaces in which thescalar field is not explicitly mentioned is to be understood to apply to bothof these cases.

6PART 1: GENERAL THEORYIf X is a vector space, Anotations will be used:cX, B c X, x E X, and A. E 1 , the followingx A {x a : a E A},x - A {x- a: a E A},A B {a b : a E A, b E B},A.A {A.a : a E A}.In particular (taking A. - 1 ), - A denotes the set of all additive inverses ofmembers of A.A word of warning : With these conventions, it may happen that 2A 'IA A (Exercise 1 ).A set Y c X is called a subspace of X if Y is itself a vector space (withrespect to the same operations, of course). One checks easily that thishappens if and only if 0 E Y and,:xY {JYcYfor all scalars ,:x and {J.A set C c X is said to be convex iftC (1 - t)CcC(0 t 1).In other words, it is required that C should contain tx (1 - t)y if x E C,y E C, and 0 t 1.A set B c X is said to be balanced if ,:xB c B for every ,:x E I withI X I LA vector space X has dimension n (dim X n) if X has a basis{ u 1 , , un} · This means that every x E X has a unique representation of theform ( X; E 1 ).x ,:x 1 u 1 ··· ,:x n unIf dim X n for some n, X is said to have finite dimension. If X {0}, thendim X 0.If X ({ (a one-dimensional vector space over the scalarfield ({}, the balanced sets are ({, the empty set 0, and every circulardisc (open or closed) centered at 0. If X R 1 (a two-dimensionalvector space over the scalar field R), there are many more balancedsets ; any line segment with midpoint at (0, 0) will do. The point isthat, in spite of the well-known and obvious identification of ({ withR 1, these two are entirely different as far as their vector space struc ture is concerned.Example. 1.5 Topological spaces A topological space is a set S in which a collec tion r of subsets (called open sets) has been specified, with the following

CHAPTER1:TOPOLOGICAL VECTOR SPACES7properties : S is open, 0 is open, the intersection of any two open sets isopen, and the union of every collection of open sets is open. Such a collec tion r is called a topology on S. When clarity seems to demand it, the topo logical space corresponding to the topology r will be written (S, r) ratherthan S.Here is some of the standard vocabulary that will be used, if S and rare as above.A set E c S is closed if and only if its complement is open. The closureE of E is the intersection of all closed sets that contain E. The interior Eo ofE is the union of all open sets that are subsets of E. A neighborhood of apoint p E S is any open set that contains p. (S, r) is a Hausdorff space, and ris a Hausdorff topology, if distinct points of S have disjoint neighborhoods.A set K c S is compact if every open cover of K has a finite subcover. Acollection r' c r is a base for r if every member of r (that is, every open set)is a union of members of r'. A collection y of neighborhoods of a pointp E S is a local base at p if every neighborhood of p contains a member of y.If E c S and if u is the collection of all intersections E n V, withV E r, then u is a topology on E, as is easily verified ; we call this the topol ogy that E inherits from S.If a topology r is induced by a metric d (see Section 1.2) we say that dand r are compatible with each other.A sequence { xn} in a Hausdorff space X converges to a point x E X(or limn oo xn x) if every neighborhood of x contains all but finitely manyof the points xn .1.6 Topological vector spacesspace X such thatSupposerts a topology on a vector(a) every point of X is a closed set, and(b) the vector space operations are continuous with respect to r.Under these conditions,ris said to be a vector topology on X, and Xis a topological vector space.Here is a more precise way of stating (a) : For every x E X, the set {x}which has x as its only member is a closed set.In many texts, (a) is omitted from the definition of a topologicalvector space. Since (a) is satisfied in almost every application, and sincemost theorems of interest require (a) in their hypotheses, it seems best toinclude it in the axioms. [Theorem 1 . 1 2 will show that (a) and (b) togetherimply that r is a Hausdorff topology.]To say that addition is continuous means, by definition, that themapping(x, y) - x y

8PART 1: GENERAL THEORYof the cartesian product X x X into X is continuous : If x; E X for i 1, 2,and if V is a neighborhood of x1 x 2 , there should exist neighborhoods v;of x; such thatSimilarly, the assumption that scalar multiplication ts continuous meansthat the mapping(a:, x) - a:xof I x X into X is continuous : If x E X, a: is a scalar, and V is a neighbor hood of a:x, then for some r 0 and some neighborhood W of x we have{3W c V whenever I {3- a: I r.A subset E of a topological vector space is said to be bounded if toevery neighborhood V of 0 in X corresponds a number s 0 such thatE c t V for every t s.Invariance Let X be a topological vector space. Associate to eacha E X and to each scalar A. # 0 the translation operator T;. and the multipli cation operator M;., by the formulas1.7T;.(x) a x,M;.(x) A.x(x E X).The following simple proposition is very important:Proposition. Taand M;. are homeomorphisms of X onto X.The vector space axioms alone imply that T;. and M;. areone-to-one, that they map X onto X, and that their inverses are T aand M 11;., respectively. The assumed continuity of the vector spaceoperations implies that these four mappings are continuous. Henceeach of them is a homeomorphism (a continuous mapping whoseinverse is also continuous).////PROOF.One consequence of this proposition is that every vector topology r istranslation-invariant (or simply invariant, for brevity) : A set E c X is open ifand only if each of its translates a E is open. Thus r is completely deter mined by any local base.In the vector space context, the term local base will always mean alocal base at 0. A local base of a topological vector space X is thus acollection !?I of neighborhoods of 0 such that every neighborhood of 0 con tains a member of ?1. The open sets of X are then precisely those that areunions of translates of members of ?1.

CHAPTER1:TOPOLOGICAL VECTOR SPACES9A metric d on a vector space X will be called invariant ifd(x z, y z) d(x, y)for all x, y, z in X.1.8 Types of topological vector spaces In the following definitions, Xalways denotes a topological vector space, with topology r.(a) X is locally convex if there is a local base(b)(c)(d)(e)(f)(g)(h)(i)!?Iwhose members areconvex.X is locally bounded if 0 has a bounded neighborhood.X is locally compact if 0 has a neighborhood whose closure is compact.X is metrizable if r is compatible with some metric d.X is an F-space if its topology r is induced by a complete invariantmetric d. (Compare Section 1 .25.)X is a Frechet space if X is a locally convex F-space.X is normable if a norm exists on X such that the metric induced bythe norm is compatible with r.Normed spaces and Banach spaces have already been defined (Section1 .2).X has the Heine-Borel property if every closed and bounded subset ofX is compact.The terminology of (e) and (f) is not universally agreed upon : Insome texts, local convexity is omitted from the definition of a Frechet space,whereas others use F-space to describe what we have called Frechet space.1.9 Here is a list of some relations between these properties of a topologi cal vector space X.(a) If X is locally bounded, then X has a countable local base [part (c) ofTheorem 1 . 1 5].(b) X is metrizable if and only if X has a countable local base (Theorem1 .24).(c) X is normable if and only if X is locally convex and locally bounded(Theorem 1 . 39).(d) X has finite dimension if and only if X is locally compact (Theorems1.21, 1 .22).(e) If a locally bounded space X has the Heine-Bore! property, then X hasfinite dimension (Theorem 1 .23).

10PART 1 : GENERAL THEORYThe spaces H(n) and C'f mentioned in Section 1 . 3 are infinite dimensional Frechet spaces with the Heine-Bore! property (Sections 1 .45,1 .46). They are therefore not locally bounded, hence not normable ; theyalso show that the converse of (a) is false.On the other hand, there exist locally bounded F-spaces that are notlocally convex (Section 1 .47).Separation PropertiesSuppose K and C are subsets of a topological vector spaceX, K is compact, C is closed, and K n C 0. Then 0 has a neighborhood Vsuch that(K V) n (C V) 0.1.10TheoremNote that K V is a union of translates x V of V (x E K). ThusK V is an open set that contains K. The theorem thus implies the exis tence of disjoint open sets that contain K and C, respectively.We begin with the following proposition, which will be usefulin other contexts as well :PROOF.If W is a neighborhood of O in X, then there is a neighborhood Uof 0 which is symmetric (in the sense that U U) and which satisfiesU U c W.-To see this, note that 0 0 0, that addition is continuous, andthat 0 therefore has neighborhoods V" V1 such that V1 V1 c W. IfU V, n V2 n ( V,) n ( V2),--then U has the required properties.The proposition can now be applied to U in place of W andyields a new symmetric neighborhood U of 0 such thatV V V V c W.It is clear how this can be continued.If K 0, then K V 0, and the conclusion of the theoremis obvious. We therefore assume that K # 0, and consider a pointx E K. Since C is closed, since x is not in C, and since the topology ofX is invariant under translations, the preceding proposition showsthat 0 has a symmetric neighborhood V, such that x V, Vx Vxdoes not intersect C; the symmetry of Vx shows then that(1)

CHAPTER1:TOPOLOGICAL VECTOR SPACES11Since K is compact, there are finitely many points Xu . , xn in K suchthat.Put V Vx,nK c (x 1 V,)Then· · · n Vx u · · · u.(xn V,J.nU (x ; Vx,i 1n V) c U (x ; V, , Vx),K Vci 1and no term in this last union intersects C V, by (1). This completesthe proof.////Since C V is open, it is even true that the closure of K V does notintersect C V; in particular, the closure of K V does not intersect C.The following special case of this, obtained by taking K {0}, is of con siderable interest. If !!I is a local base for a topological vector space X, thenevery member of :11 contains the closure of some member of f!l.1.1 1TheoremSo far we have not used the assumption that every point of X is aclosed set. We now use it and apply Theorem 1 . 1 0 to a pair of distinctpoints in place of K and C. The conclusion is that these points have disjointneighborhoods. In other words, the Hausdorff separation axiom holds :1.12TheoremEvery topological vector space is a Hausdorff space.We now derive some simple properties of closures and interiors in atopological vector space. See Section 1 .5 for the notations E and ".Observe that a point p belongs to E if and only if every neighborhood of pintersects E.1.13TheoremLet X be a topological vector space.(a) If A c X then A n (A V), where V runs through all neighborhoodsof O.(b) If A c X and B c X, then A B c A B.(c) If Y is a subspace of X, so is Y.(d) If C is a convex subset of X, so are C and C.(e) If B is a balanced subset of X, so is B; if also 0 E Bo then Bo is balanced.( f) If Eis a bounded subset of X, so is E.

12PART 1: GENERAL THEORY(a) x E A if and only if (x V) n A # 0 for every neighbor hood V of 0, and this happens if and only if x E A - V for every suchV. Since - V is a neighborhood of 0 if and only if V is one, the proofPROOF.is complete.(b) Take a E A, b E B; let W be a neighborhood of a b. Thereare neighborhoods W1 and W2 of a and b such that W1 W1 c W.There exist x E A n W1 and y E B n W2 , since a E A and b E B. Thenx y lies in (A B) n W, so that this intersection is not empty. Con sequently, a b E A B.(c) Suppose rx and fJ are scalars. By the proposition in Section1 .7, aY rxY if rx # 0; if rx 0, these two sets are obviously equal.Hence it follows from (b) thatrxY {JY rxY {JY c rxY {JY c Y ;the assumption that Y is a subspace was used in the last inclusion.The proofs that convex sets have convex closures and that bal anced sets have balanced closures are so similar to this proof of (c}that we shall omit them from (d) and (e).(d) Since co c C and C is convex, we havetC 0 (1 - t)C 0cCif 0 t 1 . The two sets on the left are open; hence so is their sum.Since every open subset of C is a subset of C o, it follows that C o isconvex.(e) If 0 I rxl 1 , then rxBo (rxB)", since x - rxx is a homeo morphism. Hence rxBo c rxB c B, since B is balanced. But rxBo is open.So rxBo c B0 If Bo contains the origin, then rxBo c Bo even for rx o,(f) Let V be a neighborhood of 0. By Theorem 1 . 1 1 , W c V forsome neighborhood W of 0. Since E is bounded, E c t W for all suffi ciently large t. For these t, we have E c tW c tV.Ill1.14TheoremIn a topological vector space X,(a) every neighborhood ofO contains a balanced neighborhood of O, and(b) every convex neighborhood of 0 contains a balanced convex neighbor hood ofO.(a) Suppose U is a neighborhood of 0 in X. Since scalar multi plication is continuous, there is a b 0 and there is a neighborhoodV of 0 in X such that rx V c U whenever I rxl b . Let W be the unionof all these sets rx V. Then W is a neighborhood of 0, W is balanced,PROOF.and W c U.

CHAPTER1 : TOPOLOGICAL VECTORSPACES13(b) Suppose U is a convex neighborhood of 0 in X. LetA n rxU, where IX ranges over the scalars of absolute value 1.Choose W as in part (a). Since W is balanced, rx - 1W W whenl rx 1 1 ; hence W c rxU. Thus W c A, which implies that the interiorAo of A is a neighborhood of 0. Clearly Ao c U. Being an intersectionof convex sets, A is convex ; hence so is Ao. To prove that A0 is aneighborhood with the desired properties, we have to show that Ao isbalanced; for this it suffices to prove that A is balanced. Choose r andfJ so that 0 r 1, I fJ I 1 . Thenr{JA n r{JrxUlal l n rrxU.lal lSince rxU is a convex set that contains 0, we have rrxUr{JA c A, which completes the proof.crxU. Thus////Theorem 1 . 1 4 can be restated in terms of local bases. Let us say that alocal base [Jl is balanced if its members are balanced sets, and let us call fJlconvex if

ABOUT THE AUTHOR In addition to Functional Analysis, Second Edition, Walter Rudin is the author of two other books: Principles of Mathematical Analysis and Real and Complex Analysis, whose widespread use is illustrated by the fact that they have been translated into a total of 13 languages.He wrote Principles of Mathematica