Real Analysis with Economic Applications - Chapter J doc

Chapter J Norm ed Linear Spaces This chapter introduces a v ery im portan t subclass of metric linear spaces, namely, the class of normed linear space s. We begin with an informal discussion that motivates the in vestigation of such sp aces. We then formalize parts of th at discussion, introduce Banach spaces, and go through a number of examples and preliminary results. The first hints of how p roductive mathematical analysis can be within the c ontext of normed linear spaces are f ound i n our fina l excursion to fixed poin t theory. Here we prove the fixed point theorems of Glick sberg, Fan, Krasnoselski˘ı and Schauder, and provide a few applications to game theory and functional equations. We then turn to the basic theory of con tin uous linear functionals definedonnormedlinearspaces,and sketch an introduction to classical linear functional analysis. Our treatment is guided b y geometric considerations for the most part, and do vetails with that of Ch apter G. In particular, w e carry our e arlier work on the Hahn-Banac h ty pe extension and separation theorems to the realm of normed linear spaces, and talk about a few fundamental results of infinite di m ensional con vex analysis, suc h as the Extreme P oin t T heorem, Krein-Milman Theorem, etc In this c hapter we a lso bring to the conclusion our work on t he classificatio n of the differences between the finite and infinite di mensional linear spaces. Finally, in order t o give at l east a glimpse of the po werful Banach space methods, we establish here the famous Uniform Boundedness Principle as a corollary of our earlier geometric findings, and go through some of its applications . The presen t t reatmen t of normed linear spaces is roughly at the same level w ith that of the classic real analysis texts by Kolmogorov and Fo m in (1970) and Roy- den ( 1994) . For a more d etailed introduction to Banac h space theory, we should recommend Kreyzig (1978), Maddox (1988) and/or the first chapter of Megginson (1998). 1 1 My coverage of linear functional analysis here is directed to wards particular applications, and is thus incomplete even at an introductory level. A more leisurely introduction would cover the open mapping and closed graph theorems, and would certainly spend some time on Hilbert spaces. I will not use these two theorems in this book, and talk about Hilbert spaces only in passing. Mo reover, my treatment com pletely ignores operator theory, which is an integral counter p art of linear functional analysis. In this regard all I can do here is to direct your atten tion to the beautiful expositions of Megginson (1998) and Schechter (2002). 450 1NormedLinearSpaces 1.1 A Geo m etr ic Motivation The Sepa rating Hyperplane T he orem attests t o the f act th at our program of marrying the m etric and linear analyses of Chapters C—G has been a successful one. Ho wever, there are still a number of shortcomings we need to deal with. Fo r instance, although w e came close in Example I.9, we hav e so far been unable to establish the existence of a nonzero con tinuous linear functional on an arbitrarily give n metric linear space. And this is for a good reason: There are Fréchet spaces on which the only continuous linear functional is the zero functional. 2 In view o f Propositions I.6 and I.7, this means that all hy perplanes in a metric linear s p ac e may be d en se, which is a truly pathologic situation. In s uch a space separation of con vex s ets is a du b ious co n cept at best. Moreover, b y the Separating Hy perplane Theorem, the only nonempty open convex subset of such a space is the entire space itself. (Why?) Let us step back for a moment and ask: What is the source of these problems? How wou ld we separate two distinct points x and y in R n b y using the Separating Hyperplane T h eorem? The answer is easy: Just take an 0 < ε <d 2 (x, y), and separate the open convex sets N ε,R n (x) and N ε,R n (y) by a closed hyper plane. So why doesn’t this argument work in an arbitrary metric linear sp ace? Because an ε-neigh borhood, while necessarily non emp ty a nd open, need not be convex in an arbitrary m etric linear space (even if this s pace is finite dimensional). This, in turn, does n ot let us in voke the Separating Hyperplane Theorem to find a closed h yperplane (and hence a nonze r o continuous function a l) at this le vel of generality. 3 So the problem lies in the fact that the ε-ne ighborhoods in a me tric linear space may not be convex. Or better, in an arbitrary metric linear space X,anε-neighborhood of 0 maynotcontainanopenconvexsetO with 0 ∈ O. Why don’t we then restrict attention to those metric linear spaces for which , for any ε > 0, there exists an open convex subset of N ε,X (0) that includes 0? Indeed, suc h a space — called a locally convex metric linear space — is free of the difficulties w e men tio ned abov e. After 2 I don’t know a simple example that w ould illustrate this. If you’re familiar with mea surable functions, you may be able to take comfort in the follo wing example: The set of all measurable real functions on [0, 1], metrized via the map (f,g) → U 1 0 min{1, |f(t) − g(t)|}dt, is a Fréchet space on which there is no nonzero continuous linear functional. (Even the linear functional f → f(0) is not con t inuous on this space. Why?) 3 The metric linear space discussed in the previous footnote provides a case in point. A simpler example is obtained b y metrizing R 2 with the metric d 1/2 (x, y):= s |x 1 − y 1 | + s |x 2 − y 2 |, x, y ∈ R 2 . (Proof. Denoting this s pace by X, check that, for any ε > 0, ( ε 2 2 , 0), (0, ε 2 2 ) ∈ N ε,X (0) but 1 2 ( ε 2 2 , 0) + 1 2 (0, ε 2 2 ) /∈ N ε,X (0) with respect to d 1/2 . Now dra w N 1,X (0) with respect to this metric.) Warning. There is a major difference between (R 2 ,d 1/2 ) and the one of the previous footnote. While the only linear functional on the latter space is the zero functional, every linear functional on the former is continuous. (Why?) The absence of “convex neighborhoods” yields a shortage of continuous linear functionals only in the case of infinite dimensional spaces. 451 all, any two dis tin ct points of a locally convex metric linear space can be sep arated b y a closed hyperplane, and hence there are plent y of continuous functionals defined on such a space. Yet, even locally convex metric linear spaces lack structure that we need for a variety o f problems that we will explore in this chapter. We want — you will see why in due course — to w ork with metric linear spaces whose ε-neighborhoods are not on ly convex but also behav e well with respect to dilations and c ontra ctio ns . Let X be a metric linear s pace. Take the open unit ball N 1,X (0), and stretc h it in your min d to dilate it at a one-to-two r atio . W ha t would you get, in tu itively? Well, there seems to be t wo obvious candidates: 2N 1,X (0) and N 2,X (0). Our Euclidean intuition suggests that these two sets should be the same, so perhaps there is no room for choice. Let’s see. Since the metric d on X is translation in variant, x ∈ N 1,X (0) implies d(2x, 0)=d(x, −x) ≤ d(x, 0)+d(0, −x)=2d(x, 0) < 2, so we have 2N 1,X (0) ⊆ N 2,X (0) indeed. Con versely, take any x ∈ N 2,X (0). Is x ∈ 2N 1,X (0)? That is to say, is 1 2 x ∈ N 1,X (0)? No, not necessarily. After all, in a metric linear space it i s possible that d(x, 0)=d( 1 2 x, 0). (In R ∞ , for instance, the distance between 0 and (2, 2, ) and that between 0 and (1, 1, ) are both equal to 1.) This is another anomaly that w e wish to avoid. Since the distance bet ween x and 0 is d(x, 0), an d 1 2 x is the midpoint of the line s e g ment bet ween 0 and x, it makes sense to require the distance between 0 and 1 2 x be 1 2 d(x, 0). That is, we wish to ha ve d( 1 2 x, 0)= 1 2 d(x, 0) for all x ∈ X. (1) In particular, this guarantees that 2N 1,X (0)=N 2,X (0). Of course, exactly the same reasoning would also justify the requiremen t d( 1 n x, 0) = 1 n d(x, 0) for ev ery n ∈ N. This, in turn , implies d(mx, 0)=md(x, 0) for all m ∈ N, and it follo ws that d(rx, 0)=rd(x, 0) for all r ∈ Q ++ . (W hy?) Conseq uent ly, since d(·, 0) is a continuous map on X,wefind that d(λx, 0)=λd(x, 0) for all λ > 0. (Yes?) Finally, if λ < 0, then, by translation in va ria nce, d(λx, 0)=d(0, −λx)= d(−λx, 0)=−λd(x, 0). In conclusion, the geometric anomaly that we wish to avoid point to wards the followin g hom o ge neity property : d(λx, 0)=|λ| d(x, 0) for all x ∈ X. (2) Metric linear spaces for which ( 2) holds pr o vides a remarkably r ic h pla yground. In particular, all ε-neigh borhoods in such a space are convex (and hence any suc h space is locally con vex). Th us, the difficulty that worried us at the beginning of our discussion does no t arise in such metric linear space s . 4 In f act, we gain a lot more b y positing (2). It turns out that the metric linear spaces with (2) prov ides an ideal environme nt to carry out a powerful convex analysis join tly w ith a functional analy sis 4 More is true: (1) alone guarantees that N ε,X (0) (and hence N ε,X (x)=x + N ε,X (0) for all x ∈ X) is a conv ex set for any ε > 0. After all, every open midpoint convex subset of a m etric linear space is convex. (Proof?) 452 of lin ear and nonlinear operators. By the time you are done with this ch apter and the next, this poin t will ha ve become abundantly clear. Exercise 1.LetX be a metric linear space. Show that every ε-neighborhood in X is con vex iff d(x, ·) is quasiconv ex for eac h x ∈ X. Exercise 2. H As noted above, a metric linear space X is called locally con vex if, for e very ε > 0, there exists an open and convex set O ε in N ε,X (0) with 0 ∈ O ε . At least one of the ε-neighborhoods in a locally conv ex metric linear space is convex. True or false? Exercise 3.Let X be a metric linear space with (1). Show that there exists a nonzero con tin uous linear functional on X. Exercise 4.LetX be a m etric linear space with (1). Sho w that X is bounded iff X = {0}. Exercise 5. In a metric linear space X with (1), would we necessarily have (2)? 1.2 Normed Linear Spaces Let us lea ve the geometric considerations we outlined above aside for a moment, and instead focus on th e following fundame nta l definition. We will return to the discussion above in about five pages. Dhilqlwlrq. Let X be a linear space. A function · : X → R + that satisfies the following properties is called a norm on X:Forallx, y ∈ X, (i) x =0if and only if x = 0, (ii) ( Absolute Hom ogeneity) λx = |λ| x for all λ ∈ R, (iii) (Subaddi tivity) x + y≤x + y . If · is a norm on X, th en we say that (X, ·) is a normed linear space.If· only s atis fies th e requireme nts (ii) and (iii), th en (X, ·) is called a seminormed linear space. 5 Recall that th e basic v iew poin t of vector calculus is to regard a “vector” x in a linear space as a directe d line segment that begins at zero and ends at x. T his allo ws 5 The idea of normed linear space was around since the 1906 dissertation of Fréchet, and a precursory analysis of it can be traced in the works of Eduard Helly and Hans Hahn prior to 1922. The m odern d efinition was given first by Stefan Banac h and Norbert Wiener, independently, in 1922. Banach t hen undertook a comprehensive analysis of such spaces whic h culminated in his ground breaking 1932 treatise. (See Dieudonné (1981).) 453 one to think of the “length” (or the “magnitude”) of a vector in a natural w a y. For instance, we think of the magnitude of a positiv e real number x as the length of the interval (0,x], and that of −x as the length of [−x, 0). Indeed, it is easily v erified that the absolute value function defines a norm on R. Similarly, it is conventional to think of the length o f a vector in R n as the distance between this vector and t he origin, and as y ou wou ld expect, x → d 2 (x, 0) defines a norm on R n . Just as the notion o f a metric generalizes the geometric notion of “distance,” therefore, the notion of a norm generalizes th a t of “length ” or “ m ag nitude” of a v ector. This interpretation also motivates the properties that a “norm” must satisfy. First, a norm m ust be nonnegativ e, because “length” is a n inheren tly nonnegative notion. Second, a nonzero vector should be assigned positiv e length, and hence propert y (i). Third, the norm o f a vector −x should equal t he norm of x, because multiplying a vector b y −1 should ch ange only the d irection of the v ector, not its length. Fourth, doubling a vector should double the norm of that vector, simply because the in tuitive notion of “len gth ” behaves in t his manner. Property (ii) of a norm is a genera liza- tion of the latter two requirements. F in ally, our Eu clidean intuition about “leng th” suggests that the norm of a vect or t hat corresponds to one side of a triangle should not exceed the sum of the norm s of the vectors that form the other two sides o f that triangle. This requirem ent is formalized as property (iii) in the f ormal definition of a norm. You m ay thin k that there may be other properties that an abstra ct notion of “leng th” should satisfy. But, as you will see in this chapter, the fram ework based on the p ro perties (i)-(iii) a lone tu r n s out to be rich eno ug h to allow for a yielding in vestig ation of the properties of linearit y and continuity in conjunction, along with a s atis fac tory geom etric analysis. When the norm under consideration is apparent from the context, it is customary to dispense with the notation (X, ·), and refer to the set X itself as a normed linea r space. We sha ll fre qu e ntly adopt this conventio n in what follows. That is, whe n we say that X is a normed linear space, you should understand that X is a linear space with a norm · lurking in the bac kground. W hen w e need to deal with tw o normed linear spaces X and Y, the norms of these spaces will be denoted by · X and · Y , respective ly. Let X be a norm ed linear space. Fo r future reference, let’s put on record an im- mediate , yet important, con s equ en c e of the a bsolu te homoge neity and subad d itivity pro pert i es of the norm on X. Take any x, y ∈ X. Note first that, by subadditivity, x = x − y + y≤x − y + y so that x−y≤x − y . Moreover, if we change t he roles of x and y i n this inequality, we get y−x≤ y − x = x − y , where the last equalit y follows from the absolute homogeneit y of ·.Thus: |x−y| ≤x − y for any x, y ∈ X. (3) 454 This inequality will be usef ul to us on sev e ral occasions, so please keep it in m in d as a companion to the subadditivity property. Exercise 6. H (Convexity of Norms) For any normed linear space X, show that x + αy−x α ≤ x + βy−x β for all x, y ∈ X and β ≥ α > 0. Exercise 7. H (Rotund Spaces ) A normed linear space X is called rotund if x + y < x + y for all linearly independent x, y ∈ X. Show that X is rotund iff   1 2 x + 1 2 y   < 1 2 x + 1 2 y for any distinct x, y ∈ X with x = y =1. (Neither  1 nor  ∞ is rotund. Right?) 1.3 Examples of Normed Linear Spaces For any n ∈ N, we can norm R n in a var iety of ways to obtain a finite dim ensional normed linear space. For any 1 ≤ p ≤∞, the p-norm · p on R n is defined as x p :=  n S i=1 |x i | p  1 p if p is finite, and x p := max{|x i | : i =1, , n} if p = ∞. Using Minkowski’s Inequality 1, it is r e ad ily chec ke d that any · p is a norm on R n for any 1 ≤ p ≤∞. (Obviously, x p = d p (x, 0) for all x ∈ R n and 1 ≤ p ≤∞.) Recall that, for an y 1 ≤ p, q ≤∞, R n,p and R n,q are “identical” metric linear spaces (in t he sense that they a re linearly homeomorphic). It w ould thus be reasonable to expect that (R n , · p ) and (R n , · q ) are “identical” in s ome formal sense a s we ll. This is in de ed the case, as we will discuss later in Section 4.2. Here are som e examples of infinite dimensional normed linear spaces. E{dpsoh 1. [1] Let 1 ≤ p<∞. The space  p becomes a normed linear space when endowed with the norm · p :  p → R + defined by (x m ) p :=  ∞ S i=1 |x i | p  1 p = d p ((x m ), 0) . (The subadd itivity of · p is equivalent to Minkow ski’s Inequality 2.) Similarly, we mak e  ∞ a normed linear space by endo w ing it with the norm · ∞ :  ∞ → R + defined by (x m ) ∞ := sup {|x m | : m ∈ N} = d ∞ ((x m ) , 0) . It is ea sily checked tha t · ∞ is indeed a norm on  ∞ . In what follo ws, when we consider  p as a n orm ed linear space, t he underlying norm is always · p , 1 ≤ p ≤∞. 455 [2] For an y nonempt y set T, the linear space B(T ) of all bounded real functions on T is no rmed by · ∞ : B(T ) → R + , where f ∞ := sup {|f(t)| : t ∈ T } = d ∞ (f,0) . For obvio us reasons, · ∞ is called the sup-norm.Ofcourse,B(N) and  ∞ are the sam e normed linear spaces . [3] Let X be a nor med linear space. By a nor m e d line a r subs p ac e Y of X, we mean a linear subs pace of X whose n orm is the restriction of the no rm of X to Y (that is, y Y := y for ea ch y ∈ Y ). Throughout t he remainder of t his book, we refer to a normed linear subspace simply as a subspace.IfY is a subspace of X and Y = X, then Y is called a proper subspace of X. As y ou would sure ly expect, giv en an y metric space T, we view CB(T ) as a subspace of B(T). Consequently, when w e talk about the no rm of an f ∈ CB(T ) (or of f ∈ C(T ) when T is compact), what we ha ve in mind is the sup-norm of this function , that is, f ∞ . Similarly, we norm c 0 , c 0 and c by the sup-norm, and hence c 0 is subspace of c 0 , c 0 is a subsp ace of c,andc is a sub s pace of  ∞ . 6 [4] For any interval I,letCB 1 (I) denote the lin ear space of all bound ed and continuo us ly differentiable real maps on I whose derivatives are bounded fu nctions on I. (Obvious ly, CB 1 ([a, b]) = C 1 [a, b].) The standard norm on CB 1 (I) is denoted by · ∞,∞ ,where f ∞,∞ := sup {|f(t)| : t ∈ I} +sup{|f  (t)| : t ∈ I} . (We leave it as an exercise to ch eck that (CB 1 (I), · ∞,∞ ) is a norm ed linear space .) From now on whenever we consider a space lik e C 1 [a, b], or more generally, CB 1 (I), as a nor me d linear space, w e will ha ve the norm · ∞,∞ in mind. [5] Let X be the linear space of all contin uou s real f u nctions on [0, 1]. Then ·∈R X + defined by f := ] 1 0 |f(t)| dt, (4) is a norm on X, but as we shall see shortly, there is a major advan tage of norming X via the sup-norm as opposed to this integral norm. [6] Let X be the linear space of all real f unctions on [0, 1] th at are continuous everywhere but finitely many points. (So, C[0, 1] ⊂ X ⊂ B[0, 1].)Then·∈R X + defined by (4) is a seminorm, but n ot a n orm , on X. [7] (Finite Prod uc t Spa ces)Takeanyn ∈ N and let X 1 , , X n be norm e d lin ear spaces. We norm the product linear space X := X n X i by using the map ·∈R X + defined by (x 1 ,x 2 , , x n ) := x 1  X 1 + ···+ x n  X n . (Is · anormonX?) 6 Reminder. c 0 is the linear space of all real sequences all but finitely many terms of which are zero; c 0 is the linear space of all real sequences that converge to 0; and c is the linear space of all conv ergent real sequences. 456 [8] (Countably Infinite P roduct Spa ces)LetX i be a normed l inear space, i = 1, 2, , and l et X := X ∞ X i . We make X a n orm ed linear space by means o f t he so-called product norm ·∈R X + whic h is defined by (x 1 ,x 2 , ) := ∞ S i=1 1 2 i min  1, x i  X i  . (Why is · anormonX?)  Exercise 8.Showthat· ∞,∞ is a norm on CB 1 (I) for any in terval I. Exercise 9. H Define ϕ :  ∞ → R + by ϕ((x m )) := lim sup |x m | . Show that ϕ is a seminorm on  ∞ . Is ϕ anormon ∞ ? Compu t e ϕ −1 (0). Exercise 10. D etermine all p in R + such that  p is a rotund normed linear space (Exercise 7). Exercise 11.Let X denote the set of all Lipschitz continuous functions on [0, 1], and define ·∈R X + by f := sup q |f(a)−f (b)| |a−b| :0≤ a<b≤ 1 r + |f(0)| . Show that (X, ·) is a normed linear space. Does f →f−|f(0)| define a norm on X? The next exercise in troduces a special class of normed linear spaces that plays a very important role in the theory of optimization. We will talk more about them later on. Exercise 12. H (Inner Product Spaces)LetX be a linear space and let φ ∈ R X×X be a function such that, for all λ ∈ R and x, x  ,y ∈ X, (i) φ(x, x) ≥ 0 and φ(x, x)=0iff x = 0, (ii) φ(x, y)=φ(y, x), (iii) φ(λx + x  ,y)=λφ(x, y)+φ(x  ,y). (We say that φ is an inner product on X, and refer to (X, φ) as an inner product space.) (a)(Cauchy-Schwarz Inequality) First, go back and read Section G.3.3, and second, prove that |φ(x, y)| 2 ≤ φ(x, x)+φ(y, y) for all x, y ∈ X. (b)Define ·∈R X by x := s φ(x, x), and show that (X,·) is a normed linear space. (Such a space is called a pre-Hilbert space.) (c)Is  2 a pre-Hilbert space? (d) Use part (b) to show that f → ( U 1 0 |f(t)| 2 dt) 1 2 defines a norm on C[0, 1]. 1.4 Metric vs. Norm ed Linea r Spaces Any m etric linear space X that has the property (2) can be con sidered as a n or med linear space under the norm · d ∈ R X + with x d := d (x, 0). 457 (Right?) Therefore the g eo me tric motivation we gav e in Section 1.1 to stu d y those metric linear spaces with (2) motivates our concentration on n ormed linear spaces. In fact, not only that the metric of any such space arises from a norm, but it is also true that there is an obvious wa y of “making” a normed linear space X into a metric linear space that satisfies (2). Indeed, any giv e n n o rm · on X readily induces a distance function d · on X in the following man ner: d · (x, y):=x − y for all x, y ∈ X. (5) (Check that d · is really a metric.) Endowing X with d · makes it a met r ic linea r space that has the property (2). To see this, take any (λ m ) ∈ R ∞ and (x m ) ∈ X ∞ with λ m → λ and d · (x m ,x) → 0 for some (λ,x) ∈ R × X. By using the subadditivity and absolute homogeneity of · , and the fact that (x m ) is a bounded real sequence, 7 we obtain d · (λ m x m , λx)=λ m x m − λx ≤λ m x m − λx m  + λx m − λx = |λ m − λ| x m  + |λ| x m − x → 0. It is obvious that d · is translation invariant and satisfies (2). This observation sho ws th at ther e is a natural wa y of v iewing a n ormed linear space as a metric linear space. For this reason, some what loosely speakin g, one often says that anormedlinearspace“is”ametriclinearspace. Throughout the rest of this book, you should always keep this viewpoint in min d . It is importan t to note that one cannot use (5) to deriv e a distance function from a norm in the absence of (2). T hat is, if (X, d) is a metric space, the function R d ∈ R X + defined by R d (x):=d(x, 0), is not necessarily a norm (even a sem inorm ) . For instance, if d is the discrete metric, then R d (x)=d( 1 2 x, 0)=R d ( 1 2 x) for all x = 0, whic h shows that R d fails to be absolutely homogeneous. For another example, notice that R d is not a seminorm on R ∞ (Exam p le I.1.[3]). In fact, since the metric of R ∞ fails to satisfy (2), it cannot possibly be induced by a norm on R ∞ . Ther efor e, w e say that a normed linear space “is” a metric linear space, but not conversely. 8 Let X be a normed line ar space. Now that we agree in view in g X as a metric spac e, nam ely (X, d · ), let us also agree to u se any notion that makes sense in the context of a metric spa ce also for X, with that notion being define d for (X, d · ). For instance, whenev er w e talk about the ε-neighbo rhood of a point x in X, we mean N ε,X (x):={y ∈ X : x − y < ε}. 7 (x m ) is bounded, because x m ≤x m − x−x for each m, and x m − x→0.Infact, more is true, no? (x m ) is a conver gent sequence. Why? (Recall (3).) 8 This does no t mean that we cannot find a norm · on R ∞ that would render (R ∞ , ρ) and (R ∞ , ρ · ) homeomorphic. Yet it is true, we cannot possibly find such a norm. (See Exercise 19.) 458 Similarly, the closed unit ball of X, w hic h w e denote henceforth as B X , ta k es the form: B X := {x ∈ X : x≤1}. Con tin uing in the s am e v ein, we declare a subset of X open iff this set is o pen in (X, d · ). By cl X (S) we mean the closure of S ⊆ X in (X, d · ), and similarly fo r int X (S) and bd X (S).Or,wesaythat(x m ) ∈ X ∞ converges to x ∈ X (we write this again as x m → x, of course) iff d · (x m ,x) → 0, that is, x m − x→0. Any topological property, along with boundedness and completeness, of a subset of X is, again, define d relative to the metr ic d · . By the same token, w e view a real function ϕ on X a s continuous, if, for any x ∈ X and ε > 0, there exists a δ > 0 (which ma y depend on both ε and x) such that |ϕ(x) − ϕ(y)| < ε for any y ∈ X with x − y < δ. Similarly, when w e talk a bout t h e cont inuity of a function Φ whic h m aps X in to another n orm e d lin ear s p ace Y, what we mean is: For all x ∈ X and ε > 0, there exists a δ > 0 such that Φ(x) − Φ(y ) Y < ε for any y ∈ X with x − y < δ. By P roposition D .1, then, a map Φ ∈ Y X is continuous iff, for any x ∈ X and (x m ) ∈ X ∞ with x m − x→0,wehaveΦ(x m ) − Φ(x) Y → 0. As an immediate example, let us ask if the norm of a norm e d linear space renders itself contin uous. A momen t’s reflectio n sho w s that not only is this the case, but the metric induced b y a norm qualify that norm as nonexpansive. Proposition 1. The norm of an y normed linear space X is a nonexpansive map on X. Proof. Apply (3).  With these definitions in mind, the findings of Chapters C —G and I apply readily to n o rm e d linear spaces. However, thanks to the absolu te hom ogen eity axiom, there is considerably more structure in a normed linear space than that contained in an arbitrary m etric linear space. This point will be clear as we proceed. Exercise 13. Let X be a normed linear space, and define the self-map Φ on X\{0} by Φ(x):= 1 x x. Show that Φ is continuous. Exercise 14.Let X be a normed linear space, and (x m ) ∈ X ∞ a Cauc hy sequence. Show that (x m ) is a convergent real sequence. Exercise 15.Prove:Asubset S of a normed linear space X is bounded iff sup{x : x ∈ S} < ∞. 459 [...]... to the realm of normed linear spaces Two major results of this sort were obtained by Julius Schauder in 1927 and 1930.15 The first of these theorems is, in fact, an immediate corollary of the Glicksberg-Fan Fixed Point Theorem The Schauder Fixed Point Theorem 1 Every continuous self-map on a nonempty compact and convex subset of a normed linear space has a fixed point.16 15 Julius Schauder (189 9-1 943)... Remark 1 One major reason why an n-dimensional linear space is a well-behaved entity is that there exists a set {x1 , , xn } in such a space X such that, for each 464 x ∈ X, there exists a unique (α1 , , αn ) in Rn with x = n αi xi (Corollary F.2) (For instance, this is the reason why every linear functional on a finite dimensional normed linear space is continuous.) The Banach spaces with Schauder bases... 1], C[0, 1]) by L(f)(t) := tf (t), and compute L ∗ Exercise 58.H Let (X, φ) be a pre-Hilbert space (Exercise 12) For any given x∗ ∈ X, ∗ define L : X → R by L(x) := φ(x, x∗ ) Show that L ∈ X ∗ and L = x∗ Exercise 59 Take any 1 ≤ p ≤ ∞, and define the right-shift and left-shift operators on p as the self-maps R and L with R(x1 , x2 , ) := (0, x1 , x2 , ) and L(x1 , x2 , ) := (x2 , x3 , ), respectively... normed linear space This result allows us to think about completeness without dealing with Cauchy sequences Proposition 3 (Banach) Let X be a normed linear space Then, X is Banach if, and only if, every absolutely convergent series in X is convergent 463 Proof Let X be a Banach space, and take any (xm ) ∈ X ∞ with For any k, l ∈ N with k > l, k d · i=1 xi , l xi k = i=1 i=1 xi − l xi = i=1 k i=l xi... were in collaboration with Jean Leray) It would be fair to consider him as one of the founders of modern nonlinear analysis 16 This result is, in fact, valid also in metric linear spaces In fact, Schauder’s original statement 469 That is, every nonempty compact and convex subset of a normed linear space has the fixed point property This is great news, to be sure Unfortunately, in applications one has... a self-map on B (b) B = C (c) ζ(B) = ζ(f (B)) Since f is ζ -condensing, it follows that ζ(B) = 0, so B is compact By the Schauder Fixed Point Theorem 1, therefore, f |B has a fixed point Exercise 44 If S is bounded, then the requirement g(S) + h(S) ⊆ S can be relaxed to (g + h)(S) ⊆ S in the statement of Krasnoselski˘ Theorem Prove this by ı’s using Sadovski˘ Fixed Point Theorem in conjunction with the... topics, including optimization theory, approximation theory, and functional equations We will later use the concepts introduced here to substantiate the convex analysis that was sketched in the earlier chapters These concepts also play a major role in Chapter K where we study the differentiation (and optimization) of nonlinear functionals 4.1 Definitions and Examples Recall that a linear functional on a metric... see this, note that, by continuity at 0, there exists a 0 < δ < 1 such that |L(y)| < 1 for all y ∈ X with y ≤ δ But then the absolute homogeneity of · entails that |L(δx)| < 1, that is, |L(x)| < 1 , for any δ x ∈ X with x ≤ 1 Thus sup {|L(x)| : x ∈ BX } < ∞ (9) A moment’s reflection shows that L being real- valued does not play a key role here If L was instead a continuous linear operator from X into another... d · (x, Y ) = x − y = α β 12 ∗ ∗ ∗ ∗ FIGURE J. 1 ABOUT HERE ∗ ∗ ∗ ∗ A famous problem of linear analysis (the so-called basis problem) was to determine if at least all separable Banach spaces have Schauder bases After remaining open for over 40 years, this problem was settled in 1973 in the negative by Per Enflo who constructed a closed subspace of C[0, 1] with no Schauder basis For a detailed account... (xm1 , xm2 , ) converges in X Thus, since a Cauchy sequence with a convergent subsequence must be convergent (Proposition C.6), we may conclude that (xm ) converges in X ∞ Exercise 28.H Let (xm ) be a sequence in a normed linear space such that xi ∞ σ(i) is absolutely convergent Show that x is also absolutely convergent for any bijective self-map σ on N ∞ Exercise 29 Let (xm ) be a sequence in a normed . Chapter J Norm ed Linear Spaces This chapter introduces a v ery im portan t subclass of metric linear spaces, namely, the class of normed linear space s. We begin with an informal. of its applications . The presen t t reatmen t of normed linear spaces is roughly at the same level w ith that of the classic real analysis texts by Kolmogorov and Fo m in (1970) and Roy- den. will explore in this chapter. We want — you will see why in due course — to w ork with metric linear spaces whose ε-neighborhoods are not on ly convex but also behav e well with respect to dilations