Real Analysis with Economic Applications - Chapter F pot

Chapter F Linear Spaces The main goal of this chapter is to provide a foundation for our subsequent introduc- tion to linear functional analysis. The latter is a vast subject, and there are many different w ays in which one can provide a first pass at it. We mostly adopt a geometric viewpoint in this book. Ind eed , we will later spend quite a bit of time covering the rudiments o f (infinite dim ensional) convex analysis. The presen t chapter in troduces the elem entary theory of linear spaces with this objective in min d . After going through a number of basic definitions and examples (where infinite dimensional spaces are given a bit more emphasis than usual), we review the notions of basis and dimension, and talk about linear operators and functionals. 1 Keeping an eye on the convex analysis to c om e , we also discuss here the n otion o f affinity at some length. In addition, we conclude an unfinished business by proving Carathéodory’s Theorem, characterize the finite dimensional linear spaces, and explore the connec- tion between hyperplanes and linear f un c tion als in some detail. On the who le, our exposition is fairly elementary, the only minor exception being the proof of the fact that every linear space has a basis — t his proof is based on the Axiom of Choice. As economic a pp lic ation s, we pro ve some basic r e sults of expected utilit y theory in the context of finite prize spaces, and introduce the elements of cooperative game theory. These applic ation s illustrate w e ll what a little linear algebra can do for y o u . 1LinearSpaces Recall that R n is naturally endow ed with three basic mathematical structures: an order structure, a metric structure and a linear structure. In the previous chapters w e ha ve studied the generalizations of the first two of these structures, whic h led to th e formu latio n of posets and metric sp aces, respectively. In this chapter we will study ho w such a generalization can be carried out in the case of the linear structu re of R n which, among other things, allo w s us to “add” an y tw o n-vect o rs. The idea is that R n is naturally equipped with an addition operation, and our immediate goal is to obtain a suitable axiomatization of this operation. 1 You can consult on any one of the numerous texts on linear algebra for more detailed treatments of these topics and related matters. My favorite is Hoffman and Kunze (1971), but this may be due to the fact that I learned this stuff from that book first. Among the more recen t and popular expositions are Broida and Williamson (1989) and Strang ( 1988). 267 1.1 Abelian Groups Let us first recall th at a binary operation • on a nonempt y set X is a map from X × X into X, but w e write x • y instead of •(x, y) for any x, y ∈ X (Section A.2.1). Dhilqlwlrq. Let X be any nonempty set, and + a binary operation on X. The doub le ton (X, +) is called a group if the follo win g four properties are satisfied . (i) (Associativity) (x + y)+z = x +(y + z) for all x, y, z ∈ X; (ii) (Existence of an identity element) There exists an element 0 ∈ X such that 0 + x = x = x + 0 for all x ∈ X; (iii) (Existence of inverse elem ents)Foreachx ∈ X, there exists an element −x ∈ X such that x + −x = 0 = −x + x. If,inaddition,wehave (iv) (Commu tativity) x + y = y + x for all x, y ∈ X, then (X, +) is said to be an Abelian (or com mutative) group. Notation. For any group (X, +), and any nonempty subsets A and B of X, we let A + B := {x + y :(x, y) ∈ A × B}. (1) In this book we w ill work e x clu sively ( an d o fte n implicitly) with Abelian g rou p s. 2 In fact, even Abelian groups do not pro vide sufficiently rich algebraic structu r e for our purposes, so w e will shortly in troduce more discipline into the picture. But w e shou ld first consid er some exam ple s of Abelian gr ou ps to make things a bit more concrete. E{dpsoh 1. [1] (Z, +), (Q, +), (R, +), (R n , +) and (R\{0}, ·) are Abelian groups where + and · are the usual addition and multiplication operations. (Note. In (R\{0}, ·) the number 1 plays t he role o f t he iden tity elemen t .) On the o ther ha nd, (R, ·) is not a group because it does not satisfy the requirement (iii). (Wha t wou ld be the inverse of 0 in (R, ·)?) Similarly, (Z, ·) and (Q, ·) are not groups. [2] (R [0,1] , +) is an Abelian grou p w h ere + is defined pointwise (i.e., f + g ∈ R [0,1] is defined by (f + g)(t):=f(t)+g(t)). [3] Let X be an y non empty set and X the class of all b ijec tive self-maps on X. Then (X , ◦) is a group , but it is not Abelian unless |X| ≤ 2. 2 “Abelian” in the term Abelian group honors the name of Niels Abel (1802-1829) who was able to make lasting contributions to group theory in his very s h ort life span. 268 [4] Let X := {x ∈ R 2 : x 2 1 + x 2 2 =1} and define x + y := (x 1 y 1 − x 2 y 2 ,x 1 y 2 + x 2 y 1 ) for an y x, y ∈ X. It is easily checked that x + y ∈ X for eac h x, y ∈ X, so this well- defines + as a binar y o peration o n X. In fact, (X, +) is an A belian group. (Verify! Hint. The iden tity element here is (1, 0).) [5] Let (X, +) be any group. The identit y element 0 of this group is uniqu e. For, if y ∈ X is another candidate for the identity elemen t, then 0+y = 0, but this implies 0 = 0 + y = y. Similarly, the inverse of any given element x is unique. Indeed, if y is an inv er se of x, then y = y + 0 = y +(x + −x)=(y + x)+−x = 0 + −x = −x. In particular, t he inverse of t he identit y element is itself. [6] (Cancellation Laws) Th e usual cancellation laws apply to any Abelian group (X, +).Forinstance,x + y = z + x iff y = z, −(−x)=x and −(x + y)=−x + −y (Section A.2.1).  Th es e examples should co nvinc e y ou that we are on the right tr ac k . The notion of Abelian group is a useful generalization of (R, +). It allows us to “add” members of arbitrary sets in conce rt with the in t uitio n supplied by R. Exercise 1.LetX be a nonempty set, and define AB := (A\B) ∪ (B\A) for an y A, B ⊆ X. Show that (2 X , ) is an Abelian group. Exercise 2. H Let (X, +) be a group, and define the binary operation + on 2 X \{∅} as in (1). What must be true for X so that (2 X \{∅}, +) is a group? Exercise 3. H Is there a binary operation on (0, 1) that would make this set an Abelian group in which the inv erse of any x ∈ (0, 1) is 1 − x? Exercise 4. H Let (X, +) be a group. If ∅ = Y ⊆ X and (Y,+) is a group (where we use the restriction of + to Y × Y, of course), then (Y,+) is called a subgroup of (X, +). (a) For any nonempty subset Y of X, show that (Y,+) is a subgroup of (X, +) iff x + −y ∈ Y for all x, y ∈ Y. (b) G ive an example to show that if (Y,+) and (Z, +) are subgroups of (X, +), then (Y ∪ Z, +) need not be a subgroup of (X, +). (c)Provethatif(Y,+), (Z, +) and (Y ∪Z, +) are subgroups of (X, +), then either Y ⊆ Z or Z ⊆ Y. Exercise 5.Let(X, +) and (Y,⊕) be two groups. A function f ∈ Y X is said to be a homomorphism from (X, +) into (Y,⊕) if f(x + x  )=f(x) ⊕ f (x  ) for all x, x  ∈ X. If there exists a bijective such map, then we say that these two groups are homomorphic. (a) Show that (R, +) and (R\{0}, · ) are homomorphic. 269 (b) Sho w that if f is a homomorphism from (X, +) into (Y,⊕),thenf(0) is the identity element of (Y,⊕), and (f(X), ⊕) is a subgroup of (Y,⊕). (c)Showthatif(X, +) and (Y,⊕) ar e homomorphic, then (X, +) is Abelian iff so is (Y,⊕). 1.2 L ine ar Sp ace s: De finition and Examples So far so good, but we are o nly half way through o ur abstraction process. We wish to hav e a generaliz ation of the linear structure of R n in a way that wou ld allo w us to algebraically represent some basic geometrical objects. For instance, the line segmen t between the vectors (1, 2) and (2, 1) in R 2 can be described algebraically as {λ(1, 2)+ (1 −λ)(2, 1) : 0 ≤ λ ≤ 1}, thanks to the lin ear structure of R 2 . By contrast, the structure of an Abelian group falls short of letting us represen t ev en suc h an elementary geometric object. This is because it is me a ningfu l to “mu lt iply” a v e ct or in a Eu clid e an sp ace w ith a real number (often called a scalar in this contex t), while there is no room for doing this with in an arbitr ary Abelian group. The next step is then to enrich the structure of Abelian groups b y defining a scalar multiplication operation on them . Once this is done, we w ill be able to describe algebraically a “line segment” in a v e ry general sense. This description w ill indeed correspond to the usual geom etric notion of line segment in the case of R 2 , and moreover , it will let us defin e and study a general notion of convexity for sets. As y ou ha ve probably already guessed, the abstract model that w e are after is none other than that of linear space . 3 Chapter G will demonstrate that this model i ndeed provides ample room for powerfu l geometric analysis. Dhilqlwlrq. Let X be a nonempty set. The list (X, +, ·) is called a linear (or vector) space if (X, +) is an A belian gro up, a n d if · is a mapping that assigns to each (λ,x) ∈ R × X an element λ · x of X (which w e denote simply as λx)suchthat, for all α, λ ∈ R and x, y ∈ X, we have (v) (Associativity) α(λx)=(αλ)x; (vi) (Distributivity) (α + λ)x = αx + λx and λ(x + y)=λx + λy; (vii) (The unit rule) 1x = x. In a linear space (X, +, ·), the mappings + and · are called addition and scalar multip licat io n operations on X, respectively. When the c ontext makes t he nature of these operations clear, we may refer to X itself as a lin ear space. The iden tity 3 We owe the first modern definition of linear space to the 1888 w ork of Giuseppe Peano. While initially ignored by the profession, the original tre atment of Peano was amaz ingly modern. I recall Peter Lax telling me once that he wo uld not be able to tell from reading certain parts of Peano’s work that it was not written instead in 1988. 270 element 0 is called the origin (or zero), and any member of X is referred to as a vector.Ifx ∈ X\{0}, then we say that x is a nonzero vector in X. Notation. Let (X, +, ·) be a linear space , and A, B ⊆ X and λ ∈ R. Then, A + B := {x + y :(x, y) ∈ A × B} and λA := {λx : x ∈ A}. For simp licity, we wr ite A + y for A + {y}, and sim ilarly, y + A := {y} + A. E{dpsoh 2. [1 ] The mos t trivial e xample of a linear spac e is a sin gleton set where the unique member of the se t is designated as the origin. Naturally, this space is denoted as {0}. Any linear space that contains more than one vector is called nontrivial. [2] Let n ∈ N. A v ery important examp le of a linear space is, of course, our beloved R n (Remark A.1). On the other hand, e ndo wed with the usual addition and scalar mu ltiplic ation operations, R n ++ is not a linear space since it does not contain the origin. This is o f co urse not the on ly problem. After all, R n + is not a linear spac e either (under the usual oper ation s), for it d oes not con tain the inverse of any no nzero vector. Is [−1, 1] n a linear space? How about {x ∈ R n : x 1 =0}? {x ∈ R n : S n x i =1}? {x ∈ R n : S n x i =0}? [3] In this book, we always thin k of the sum of two real function s f and g defined on a n on emp ty set T as the real function f + g ∈ R T with (f + g)(t):=f(t)+g(t). Similarly, the product of λ ∈ R and f ∈ R T is the function λf ∈ R T defined b y (λf)(t):=λf(t). (Recall Section A.4.1.) In particular, we consider the real sequenc e space R ∞ (which is no ne other than R N ) and the fu nctio n spaces R T and B(T) (for an y nonempty set T) a s linear spaces under these operations. Similarly,  p (for any 1 ≤ p ≤∞), along with the function spaces CB(T) an d C(T ) (for an y metric space T ), are linear spaces under these operations (why?), and when we talk about these spaces, it is these operations that w e ha ve in mind. The same goes as w ell for other function sp ac e s, such as P(T ) or the spa ce of all po lyn om ials on T of degree m ∈ Z + (for an y nonempty subset T of R). [4] Since the negative of an increasing function is decreasing, the set of a ll increas- ing real functions on R (or on any compact interv al [a, b] with a<b) is not a linear space under the usual operations. Less trivially, the set of all monotonic self-maps on R is not a linear space either. To see this, observ e that the self-map f defined on R by f(t):=sint +2t is an increasing function. (For, f  (t)=cost +2≥−1+2≥ 0 for all t ∈ R.)Buttheself-mapg defined on R by g(t):=sint − 2t is a d e creasing function , and yet f + g is ob viou sly not m onotonic. For another ex am p le, we note that the set of all semicontinuous self-map s on R does not form a linear space under the usual operations. (Wh y?)  It will become clear shortly that linear spaces provide an ideal structure for a proper inves tigation of convex sets. For the moment, however, all you need to d o is 271 to recognize that a linear space is an algebraic infrastructure relative to which the addition and s calar multip licatio n operation s behave in concert w ith intuition, tha t is, the way the corresponding operations behave on a Euclidean space. To derive this point hom e, let us d erive some preliminary f acts a bout these operations that we sh all laterinvokeinaroutinemanner. Fix a linear space (X,+, ·). First of a ll, we have λ0 = 0 for any λ ∈ R. Indeed, λ0 + λ0 = λ(0 + 0)=λ0 = 0 + λ0 so that, by Exam p le 1.[6], λ0 = 0. Asimilar reasonin g giv e s us that 0x = 0 for any x ∈ X. More generally, for an y x = 0, λx = 0 if and only if λ =0. Indeed, if λ =0and λx = 0, then x = 1 λ λx = 1 λ 0 = 0 . From this observation we deduce easily that, for any x = 0, αx = βx if and only if α = β, which, in particular, implies that any linear space other than {0} contains uncount- ably many vectors. (Exactly which properties of a linear space did we use to get this conclusion?) Finally, let us show that we ha ve −(λx)=(−λ)x for all (λ,x) ∈ R × X. (2) (Here −(λx) is the inverse of λx, while (−λ)x is the “product” of −λ and x; so the claim i s not trivia l.) Indeed , 0 =0x =(λ −λ)x = λx+(−λ)x for an y (λ,x) ∈ R ×X, andifweadd−(λx) to both sides of 0 = λx +(−λ)x, we find −(λx)=(−λ)x. Than ks to (2), there is no difference bet ween (−1)x and −x, and hence bet ween x +(−1)y and x + −y. In what follows, we shall write x − y for e ithe r of the latter exp ressions since we are now confiden t that there is no room for confusio n. Exercise 6.Define the binary operations + 1 and + 2 on R 2 by x+ 1 y := (x 1 +y 1 ,x 2 + y 2 ) and x + 2 y := (x 1 + y 1 , 0). Is (R 2 , + i , ·), where · maps eac h (λ,x) ∈ R × X to (λx 1 , λx 2 ) ∈ R 2 , a linear space, i =1, 2? What if · maps each (λ,x) to (λx 1 ,x 2 )? What if · maps each (λ,x) to (λx 1 , 0)? Exercise 7.Let(X, +, ·) be a linear space, and x, y ∈ X. Show that ({αx + λy : α, λ ∈ R},+, ·) is a linear space. Henceforth, we use the notation X inst ead of (X, +, ·) foralinearspace,but you sh ou ld always kee p in mind that w h at makes a linear s p a c e “linear” is the two operations defined on it. Two different t ypes of a dd ition and scalar mu ltiplication operations on a given set ma y well endow this set with different l inear structures, and henc e yield two v e r y differen t linear space s. 272 1.3 Linear Subspaces, Affine Manifolds and Hyperplanes One method of o btaining other linear spaces from a given linear space X is to consider those s ub sets of X which are them selv es linear spaces under the inherited operations. Dhilqlwlrq. Let X be a linear space and ∅ = Y ⊆ X. If Y is a linea r sp ac e w ith the same operations of addition and scalar multiplication as with X, then it is called a linear s ubspace of X. 4 If, further, Y = X, then Y is called a proper linear subspace of X. Th e following exerc ise p r ovide s an alterna tive (and of course equ ivalent) definition of the notion of linear subsp ace . We w ill use this alternative formulation free ly in what follows. Exercise 8 . Let X be a linear space and ∅ = Y ⊆ X. Show that Y is a linear subspace of X iff λx + y ∈ Y for each λ ∈ R and x, y ∈ Y. E{dpsoh 3. [1] [0, 1] is not a linear subspace of R whereas {x ∈ R 2 : x 1 + x 2 =0} is a proper linear subspace of R 2 . [2] For any n ∈ N, R n×n is a linear space under the us ua l matrix operations of addition and s ca lar multiplication (Remark A .1). Th e set of all symmetric n × n matrices, that is, {[a ij ] n×n : a ij = a ji for each i, j} is a linear subspace of t his space. [3] For any n ∈ N and linear f : R n → R (Section D.5.3), {x ∈ R n : f(x)=0} is a line ar subspace of R n . Would this conclusion be true if all w e knew was that f is additive? [4] For any m ∈ N, the set of constant functions on [0, 1] is a p roper linear subspace of the set of all polynomials on [0, 1] of degree at most m. The latter set is a proper linear subspace of P[0, 1] whic h is a proper linear subspace of C[0, 1] whic h is itself a pro per linear subsp a ce of B[0, 1]. Finally, B[0, 1] is a proper linear subspa ce of R [0,1] .  Exercise 9 . (a)Is 1 a linear subspace of  ∞ ? (b)Isthesetc of all convergent real sequences a linear subspace of  ∞ ? Of  1 ? (c)Letc 0 be the s et of all real sequences all but finitely man y terms of which are zero. Is c 0 a linear space (under the usual (pointwise) operations). Is it a linear subspace of c? (d)Is{(x m ) ∈ c 0 : x i =1for some i} alinearsubspaceofc 0 ? Of R ∞ ? 4 Put more precisely, if the addition and scalar multiplication operations on X are + and ·, respectively, then by a linear subspace Y of X, we mean the linear space (Y,⊕, ), where Y is a nonempty subset of X, and ⊕ : Y 2 → Y is the restriction of + to Y 2 and  : R × Y → Y is the restriction of · to R × Y. 273 Exercise 10 . Show that the intersection of any collection of linear subspaces of a linear space is itself a linear subspace of that space . Exercise 11 .IfZ is a linear subspace of Y and Y a linear subspace of the linear space X, is Z necessarily a linear subspace of X? Exercise 12. H Let Y and Z be linear subspaces of a linear space X. Prov e: (a) Y + Z is a linear subspace of X; (b)IfY ∪ Z is a linear subspace of X, then either Y ⊆ Z or Z ⊆ Y. Clearly, {x ∈ R 2 : x 1 + x 2 =1} is not a linear subspace of R 2 .(Thissetdoes not even conta in the origin of R 2 . And y e s, this is crucial!) O n the other hand, geometrically speaking, this set is very “similar” to the linear subspace {x ∈ R 2 : x 1 + x 2 =0}. Indeed, the latter is nothing but a parallel shift (translation)ofthe former set. It is thus n ot surpris in g that such sets pla y an important role in geome tric applications of linear algeb ra. They certa inly deserve a nam e . Dhilqlwlrq. A subset S of a linear space X is said to be an affine manifold of X if S = Z + x ∗ for some linear subspace Z of X and some vector x ∗ ∈ X. 5 If Z is a ⊇-maximal proper linear subspace of X, S is then called a hyperplane in X. (Equivalently, a hy perplane is a ⊇-maximal proper affine manifold.) As for ex amples, no te that there i s no hyperplane in the trivial sp ace {0}. Since the only proper linear subspace o f R is {0}, any one-point set in R and R itself a re the only a ffine m anifolds i n R. So, all hyperplanes in R are s ing leton se ts. In R 2 , any one- point set, an y line (with no endpoin ts) and the entire R 2 are the only affine mani folds. A hyperplane in this space is necessarily of the f orm {x ∈ R 2 : a 1 x 1 + a 2 x 2 = b} for some real numbers a 1 ,a 2 with at least one of them being nonzero, and some real number b. Finally, a ll hy perplanes are of th e form of ( infinitely ex tending) p lanes in R 3 . A good wa y of thin kin g intuitively about the n otion of affinity in linear analysis is this: affinity=linearity+translation. Since we think of 0 as the origin of the linear space X — this is a geometric inter- pretation; don’t forget that the definition of 0 is purely algebraic — it m akes sense to view a linear s u bspace of X as untranslated (relative to t he origin of the sp ace), for a linear subspa ce “passes through” 0. The following simple but important observa tion th u s gives support to our informal equation above: An affine manifold S of a linear space X is a linear s ubs pace of X iff 0 ∈ S. (Proof.IfS = Z + x ∗ for some linear subspace Z of X and x ∗ ∈ X, then 0 ∈ S implies −x ∗ ∈ Z. (Yes?) Since Z is a linear space, we then have x ∗ ∈ Z, and hence Z = Z + x ∗ = S.) An immediate corollary of 5 Reminder. Z + x ∗ := {z + x ∗ : z ∈ Z}. 274 this is: If S is an affine manifold of X, then S − x is a linear subspace of X for any x ∈ S. Moreover, this subspace is determined independen tly of x,becauseif S is an affine manifold, then S − x = S − y for any x, y ∈ S. (3) We leave the proof of t his assertion as an easy exercise. 6 The following result, a counterpart of Exercise 8, p rovides a useful characterization of affine manifolds. Proposition 1. Let X be a linear space and ∅ = S ⊆ X. Then S is an affine manifold of X if, and only if, λx +(1− λ)y ∈ S for any x, y ∈ S and λ ∈ R. (4) Proof. If S = Z + x ∗ for some linear subspace Z of X and x ∗ ∈ X, then for an y x, y ∈ S there exist z x and z y in Z such that x = z x + x ∗ and y = z y + x ∗ . It follo ws that λx +(1− λ)y =(λz x +(1− λ)z y )+x ∗ ∈ Z + x ∗ for any x, y ∈ S and λ ∈ R. Conversely, a ssume that S sat i sfies (4). Pic k an y x ∗ ∈ S and define Z := S − x ∗ . Then S = Z +x ∗ , so we will be done if w e can sho w that Z is a linear subspace of X. Than ks to Exercise 8, all we need to do is, then, to establish that Z is closed under scalar mu ltip lication a n d ve ctor addition. To pro ve th e former fact, notice that if z ∈ Z then z = x − x ∗ for some x ∈ S, so, by (4), λz = λx − λx ∗ =(λx +(1− λ)x ∗ ) − x ∗ ∈ S − x ∗ = Z for any λ ∈ R. To prove the latter fact, tak e any z, w ∈ Z, and note that z = x − x ∗ and w = y − x ∗ for some x,y ∈ S. B y (4), 2x − x ∗ ∈ S and 2y − x ∗ ∈ S. Theref ore, applyin g (4) again, z + w = 1 2 (2x − x ∗ )+ 1 2 (2y − x ∗ ) − x ∗ ∈ S − x ∗ = Z, and the proof i s complete.  The geometric nature of affine manifolds and h yperplanes mak es them indispens- able tools for co nvex analysis, and we will indeed u s e them exten sively in what follows . For the time being , however, w e lea ve their discussion at this primitive stage, and instead press on reviewing the fundamental concepts of linear a lgebra. We w ill revisit these n otions at various points in the subsequent developmen t. 6 Just to fix the intuition, you m ight want to verify everything I said in this p aragraph in the special case of the affine manifolds {x ∈ R 2 : x 1 + x 2 =1}, {x ∈ R 3 : x 1 + x 2 =1} and {x ∈ R 3 : x 1 + x 2 + x 3 =1}. (In particular, draw the pictures of these manifolds, and see how they are obtained as translations of specific linear subspaces.) 275 1.4 Span and Affine Hull of a Set Let X be a linear space and (x m ) ∈ X ∞ . We define m S i=1 x i := x 1 + ···+ x m and m S i=k x i := m−k+1 S i=1 x i+k−1 for any m ∈ N and k ∈ {1, , m}, and note that there is no am b igu ity in this definition, thanks to the associativity of the addition operation + on X. (By the same token, for any nonempty finite subset of T of X, it is without ambiguity to write S x∈T x for the sum o f all me mbers of T ). As in the case of sums of real n um bers, w e will write S m x i for S m i=1 x i in the text. Dhilqlwlrq. For any m ∈ N, by a linear combination of the v ectors x 1 , , x m in a linear space X, we mean a v ector S m λ i x i ∈ X, where λ 1 , , λ m are an y real n umbers (called the coefficien ts of the linear com bination). If, in addition, w e have S m λ i =1, then S m λ i x i is referred to as an affine combin ation of the vectors x 1 , , x m . If, on the other hand, λ i ≥ 0 for each i, then S m λ i x i is called a positive linear combinatio n of x 1 , , x m . Finally, if λ i ≥ 0 for each i and S m λ i =1, then S m λ i x i is called a convex com b in ation of x 1 , , x m . Equivalently, a linear (affine (convex)) combination of the elements o f a nonempty finite subset T of X is S x∈T λ(x)x, where λ ∈ R T (and S x∈T λ(x)=1(and λ(T ) ⊆ R + )). Dhilqlwlrq. The set of all linear c ombinations of finitely man y mem bers of a non- empty subset S of a linear space X is called the span of S (in X), and is d enoted by spa n(S). That is, for an y ∅ = S ⊆ X, span(S):=  m S i=1 λ i x i : m ∈ N and (x i , λ i ) ∈ S × R,i=1, , m  , or equivalen tly, span(S)=  S x∈T λ(x)x : T ∈ P(S) and λ ∈ R T  , where P(S) is the class of all nonempty finite subsets of S. By con vention, we let span(∅)={0}. Giv e n a linear space X, span({0})={0} and span(X)=X while span({x})= {λx : λ ∈ R}. The span of a set i n a linear space is always a lin e ar space (ye s?), and it is thus a linear subspace of the moth e r space . What is more, span(S) is the smallest (i.e. ⊇-minimum) linear subspace of the mother space that con tains S. Especially when S is finite, this linear subspace has a very concrete description in that every v ector in it can be expressed as linear combination of all the v ectors in S. (Why?) 276 [...]... which is distinct from the standard basis 282 [3] Fix m ∈ N, and let X denote the set of all polynomials on R which are of degree at most m If fi ∈ R[0,1] is defined by fi (t) := ti for each i = 0, , m, then {f0 , , fm } is a basis for X (Note Linear independence of {f0 , , fm } follows from the fact that any polynomial of degree k ∈ N has at most k roots.) We have dim(X) = m + 1 [4] Let e1 := (1, 0, 0,... prompts the introduction of the following important concept D A basis for a linear space X is a ⊇-minimal subset of X that spans X That is, S is a basis for X if, and only if, (i) X = span(S); and (ii) If X = span(T ), then T ⊂ S is false 10 There are various refinements of Carathéodory’s Theorem that posit further structure on S and provide better bounds for the number of elements of S that is needed to... Helly’s Intersection Theorem, one of the most famous results of convex analysis Exercise 18.H Let n ∈ N Prove: (a) (Helly’s Intersection Theorem) If S is a finite class of convex subsets of Rn such that |S| ≥ n + 1, and T = ∅ for any T ⊆ S with |T | = n + 1, then S = ∅ (b) If S is a class of compact and convex subsets of Rn such that |S| ≥ n + 1 and T = ∅ for any T ⊆ S with |T | = n + 1, then S = ∅ ∗ 1.6... S, except that f( s) = 0 and f (s ) = 1 for some s, s ∈ S (We have |S| ≥ 2, right?) Show that f( x + y) = f (x) + f( y) for all x, y ∈ R, but there is no α ∈ R such that f (x) = αx for all x ∈ R Conclusion: Cauchy’s Functional Equation admits nonlinear solutions (Compare with Lemma D.2) ∗ 2 Exercise 25.16 Prove: If A and B are bases for a linear space, then A ∼card B Linear Operators and Functionals 2.1... the function KL ∈ RX by KL(x) := K(x)L(x) Prove or disprove: KL is a linear functional on X iff it is the zero functional on X Exercise 29 Let X be the linear space of all polynomials on R of degree at most 1 Define the linear functionals K and L on X by K (f) := f( 0) and L (f ) := f (1), 288 respectively Show that {K, L} is a basis for L(X, R) How would you express L1 , L2 ∈ L(X, R) with 1 L1 (f ) := f( t)dt... αn such that L(x) = for all x ∈ Rn Since L|S = ϕ, therefore, ϕ(x) = n αi xi for all x ∈ S n αi xi In what follows we will obtain a similar characterization of affine functions First, let us define these functions properly D Let S be a nonempty subset of a linear space X, and denote by P(S) the class of all nonempty finite subsets of S A real map ϕ ∈ RS is called affine if ϕ λ(x)x = x∈A for any A ∈ P(S) and... applying the affineness of + λ s 1+λ again The converse claim Proof of Proposition 7 The “if” part is easy and left as an exercise To prove the “only if” part, take any complete preference relation on S that satisfies the hypotheses of the assertion, and assume that p∗ p∗ (If p∗ ∼ p∗ , the proof would be completed upon choosing u to be any constant function Verify!) Then, by Lemma 2, for any α, β ∈ [0, 1],... next part of the exercise points to a very surprising implication of this observation (and hence of Zorn’s Lemma) (b) Let S be a Hamel basis for R Then, for every nonzero x ∈ R, there exist a unique mx ∈ N, a unique subset {s1 (x), , smx (x)} of S, and a unique subset {q 1 (x), , qmx (x)} of Q\{0} such that x = mx qi (x)si (x) Define the self-map f mx on R by f( x) := qi (x )f (si (x)), where f is defined... proof Equipped with the notion of affine independence, we are now ready to settle that score E 4 (Proof of Carathéodory’s Theorem) Let us first recall what Carathéodory’s Theorem is about Fix any n ∈ N and recall that, for any subset A of Rn , co(A) corresponds to the set of all convex combinations of finitely many members of A Carathéodory’s Theorem states the following: Given any nonempty S ⊆ Rn , for... a12 )t2 for all t ∈ R (a) Show that L ∈ L(R2 , Y ) (b) Compute null(L) and find a basis for it (c) Compute L(R2×2 ) and find a basis for it Exercise 27 Let X denote the set of all polynomials on R Define the self-maps D and L on X by D (f) := f and L (f )(t) := tf (t) for all real t, respectively Show that D and L are linear operators and compute D ◦ L − L ◦ D Exercise 28.H Let K and L be two linear functionals . thin k of the sum of two real function s f and g defined on a n on emp ty set T as the real function f + g ∈ R T with (f + g)(t): =f( t)+g(t). Similarly, the product of λ ∈ R and f ∈ R T is the function. that the self-map f defined on R by f( t):=sint +2t is an increasing function. (For, f  (t)=cost +2≥−1+2≥ 0 for all t ∈ R.)Buttheself-mapg defined on R by g(t):=sint − 2t is a d e creasing function. necessarily of the f orm {x ∈ R 2 : a 1 x 1 + a 2 x 2 = b} for some real numbers a 1 ,a 2 with at least one of them being nonzero, and some real number b. Finally, a ll hy perplanes are of th e form of