THEOREMS, ETC., FOR MATH 516

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1 THEOREMS, ETC., FOR MATH 516 Results labeled Theorem Ea.b.c (or Proposition Ea.b.c, etc.) refer to Theorem c from section a.b of Evans book (Partial Differential Equations). Proposition 1 (=Proposition 10.2). (a) A bounded linear operator is uniformly continuous. (b) A linear operator which is continuous at one point is bounded. Proposition 2 (=Proposition 10.3). The set of all bounded linear operators from a normed linear space to a Banach space is also a Banach space. Theorem 3 (=Theorem 10.4, Hahn-Banach). Let X be a vector space, let p be a real- valued function on X such that (x + y) p(x) + p(y) and p(αx) = αp(x) for all x and y in X and all α R. Suppose f is a linear functional on a subspace S of X such that f(s) p(s) for all s S. Then there is a linear functional F on X which agrees with f on S such that F (x) p(x) for all x X. Proposition 4 (=Proposition 10.5). Let X be a vector space, let p be a real-valued function on X such that (x + y) p(x) + p(y) and p(αx) = αp(x) for all x and y in X and all α R. Suppose f is a linear functional on a subspace S of X such that f(s) p(s) for all s S. Let G be an Abelian semigroup of linear operators on X and suppose that p(ax) p(x) for all x X and f(as) = f(s) for all s S. Then there is a linear functional F on X which agrees with f on S such that F (x) p(x) and F (Ax) = F (x) for all x X. Proposition 5 (=Proposition 10.6). Let X be a normed linear space and let x be a nonzero element of X. Then there is a bounded linear functional f on X such that f(x) = f x. Proposition 6 (=Proposition 10.7). Let X be a normed linear space, let T be a subspace of X and let y be an element of X such that d(y, T ) δ for some positive δ. Then there is a bounded linear functional f on X such that f(y) = δ, f 1, and f = 0 on T. Lemma 7 (=Lemma 10.9). Let X and Y be Banach spaces and let A be a bounded linear transformation of X onto Y. Then the image of the unit ball of X contains an open ball of Y with center 0. Theorem 8 (=Proposition 10.10, Open Mapping Theorem). Let A be a bounded linear transformation from a Banach space X onto a Banach space Y. Then A is an open mapping. If A is also one-to-one, then A is an isomorphism. Proposition 9 (=Proposition 10.11). Let X be a vector space, let 1 and 2 be two norms on X and suppose that (X, 1 ) and (X, 2 ) are both Banach spaces. If there is a constant C such that x 1 C x 2 for all x X, then there is a constant K such that x 2 K x 1 1

2 2 THEOREMS, ETC., FOR MATH 516 for all x X. Theorem 10 (=Theorem 10.12, Closed Graph Theorem). Let X and Y be Banach spaces and let A: X Y be a linear transformation. If, whenever x n is a convergent sequence in X with limit x such that Ax n is a convergent sequence in Y with limit y, we have y = Ax. Then A is bounded. Proposition 11 (=Proposition 10.13, Uniform Boundedness Principle). Let X be a Banach space, let Y be a normed linear space, and let F be a family of linear operators from X to Y. Suppose that, for every x X, there is a number M x such that T x M x for all T F. Then there is a constant M such that T M for all T F. Proposition 12 (=Proposition 8.5). A collection B of subsets of a set X is a base for a topology on X if and only if (i) For every x X, there is B B such that x B. (ii) For every x X and every B 1 and B 2 in B such that x B 1 B 2, there is B 3 B such that x B 3 B 1 B 2. Proposition 13 (=Proposition 8.6). A topological space satisfies T 1 if and only if points are closed. Lemma 14 (=Lemma 8.7, Urysohn s Lemma). Let A and B be disjoint closed subsets of a normal space X. Then there is a continuous, real-valued function f on X such that 0 f 1 on X, f 0 on A, and f 1 on B. Theorem 15 (=Theorem 8.8, Tietze s Extension Theorem). Let A be a closed subset of a normal space X and let f : A R be continuous. Then there is a continuous function g : X R such that g = f on A. Proposition 16 (=Proposition 10.14). Let X be a vector space. (a) Let T be a topology on X which makes X a topological vector space. Then there is a local base B for T such that (i) If U and V are in B, then there is W B such that W U V. (ii) If U B and x X, then there is V B such that x + V U. (iii) If U B, then there is v B such that V + V U. (iv) If U B and x X, then there is α R such that x αu. (v) If U B and if α [ 1, 1] is nonzero, then αu U and αu B. (b) Conversely, if B is a collection of subsets of X which all contain 0 which satisfies properties (i) (v), then B is a local base for a topology which makes X into a topological vector space. This topology is Hausdorff if and only if (vi) U = {0}. U B Proposition 17 (=Proposition 10.15, Tychonoff). Let X be a finite-dimensional, Hausdorff topological vector space. Then X is isomorphic to R n for some n. Proposition 18 (=Proposition 10.16). A subspace of a topological vector space is closed if and only if it s weakly closed. Theorem 19 (=Theorem 10.17, Alaoglu). S = {f : f 1} is weak compact in X. Lemma 20 (=Lemma 10.18). If K 1 and K 2 are convex, then so are K 1 K 2, λk 1 (for any real λ), and K 1 + K 2.

3 THEOREMS, ETC., FOR MATH Lemma 21 (=Lemma 10.19). If 0 is an internal point of a convex set K, then the support function p satisfies the following conditions: (i) p is positively homogeneous. (ii) p is subadditive. (iii) {x : p(x) < 1} K {x : p(x) 1}. Theorem 22 (=Theorem 10.20). If K 1 and K 2 are disjoint convex set in a vector space X and if one of them has an internal point, then there is a nonzero linear functional which separates them. Proposition 23 (=Proposition 10.21). Let X be a vector space. (a) Let N be a family of convex subsets of X containing 0 and suppose that they satisfy the following conditions: (i) if N N, then each point of N is an internal point. (ii) For N 1 and N 2 in N, there is N 3 N such that N 3 N 1 N 2. (iii) If N N and 0 < α 1, then αn N. Then N is a local base for a topology on X which makes X a locally convex topological vector space. (b) Conversely, if X is a locally convex topological vector space, then there is a local base N satisfying (i) (iii). Proposition 24 (=Proposition 10.22). Let F be a closed convex subset of a locally convex topological vector space X and let x X K. Then there is a continuous linear functional f on X such that f(x) < inf{f(y) : y K}. Corollary 25 (=Corollary 10.23). A convex subset of a locally convex topological vector space is strongly closed if and only if it s weakly closed. Corollary 26 (=Corollary 10.24). If x and y are distinct points of a locally convex, Hausdorff topological vector space, then there is a continuous linear functional such that f(x) f(y). Lemma 27 (=Lemma 10.25). Let f be a continuous linear functional defined on a convex subset K of a topological vector space. Then the set of points on which f attains its maximum is a supporting set of K. Theorem 28 (=Theorem 10.26, Krein-Milman). Let K be a compact, convex subset of a locally convex, Hausdorff topological vector space. Then K is the closed convex hull of its extreme points. Proposition 29 (=Proposition 9.2, second part). A compact subset of a Hausdorff space is closed. Proposition 30 (=Proposition 9.5). A one-to-one continuous function from a compact space onto a Hausdorff space is a homeomorphism. Proposition 31 (=Proposition 9.6). The continuous image of a countably compact space is countably compact. Proposition 32 (=Proposition 9.7). A space is countably compact if and only if it has the Bolzano-Weierstrass property.

4 4 THEOREMS, ETC., FOR MATH 516 Proposition 33 (=Proposition 9.8). A sequentially compact space is countably compact. Every first countable, countably compact space is sequentially compact. Proposition 34 (=Proposition 9.10). An upper semicontinuous real-valued function on a countably compact space is bounded above and attains its maximum value. Corollary 35 (=Proposition 9.9). A continuous real-valued function on a countably compact space is bounded above and below, and it attains its maximum and minimum values. Proposition 36 (=Proposition 9.11, Dini). Let f n be a sequence of upper semicontinuous real-valued functions on a countably compact space X such that f n (x) decreases to 0 for each x X. Then f n 0 uniformly on X. Lemma 37 (=Lemma 9.12). Let A be a collection of subsets of a set X with the finite intersection property. Then there is a collection B of subsets of X with the finite intersection property which contains A and which is maximal. Lemma 38 (=Lemma 9.13). Let B be a maximal collection of subsets of a set X with the finite intersection property. (i) Any intersection of finitely many elements of B is an element of B. (ii) If C X and B C for all B B, then C B. Theorem 39 (=Theorem 9.14, Tychonoff). Any product of compact spaces is compact. Proposition 40 (=Proposition 9.15). Let K be a compact subset of a locally compact Hausdorff space X. (i) Then there is an open set U such that K U and U is compact. (ii) For any such U, there is a continuous nonnegative, real-valued function f such that f = 0 on Ũ and f = 1 on K. If K is a G δ set, then this function also has the property that f < 1 on K. Proposition 41 (=Proposition 9.16). Let {O λ } be an open cover of a compact subset K of a locally Hausdorff space X. Then there is a finite collection {ϕ 1,..., ϕ n } of continuous, nonnegative, real-value functions subordinate to {O λ } such that ϕ ϕ n = 1 on K. Proposition 42 (=Proposition 9.29). Let L be a lattice in C(X) for some compact Hausdorff space X. If h, defined by h(x) = inf f L f(x), is continuous, then, for any ε > 0, there is a function g L such that 0 g h ε. Lemma 43 (=Lemma 9.31). Let L be a family of real-valued functions on a set X such that (i) L separates points. (ii) If f L and c R, then f + c and cf are in L. Then for any a and b in R and any x y in X, there is f L such that f(x) = a and f(y) = b. Lemma 44 (=Lemma 9.32). Let L be a lattice in C(X) for some compact space X and suppose L satisfies properties (i) and (ii) from Lemma 43. Let p X and let F be a closed subset of X with p / F. Then for any real numbers a b, there is f L with f a in X, f > b in K and f(p) = a.

5 THEOREMS, ETC., FOR MATH Proposition 45 (=Proposition 9.30). Let L be a lattice in C(X) for some compact space X and suppose L satisfies properties (i) and (ii) from Lemma 43. Then L = C(X). Lemma 46 (=Lemma 9.33). For any ε > 0, there is a polynomial P ε such that t P ε (t) < ε for t [ 1, 1]. Theorem 47 (=Theorem 9.34, Stone-Weierstrass). Let X be a compact space and let A be an algebra in C(X) that contains the constant functions and separates points. Then A = C(X). Theorem 48 (=Corollary 9.35, Weierstrass Theorem). Every continuous function on a closed bounded set of R n can be uniformly approximated by polynomials. Proposition 49 (=Proposition 11.17). Let (X, B) be a measurable space, let µ n be a sequence of measures on (X, B) that converge setwise to a measure µ, and let f n be a sequence of nonnegative measurable functions that converge pointwise to a function f. Then f dµ lim f n dµ n. Proposition 50 (=Proposition 11.18). Let (X, B) be a measurable space, let µ n be a sequence of measures on (X, B) that converge setwise to a measure µ, let g n be a sequence of nonnegative measurable functions that converge pointwise to a integrable function g, and let f n be a sequence of measurable functions that converge pointwise to a function f. Suppose also that f n g and that lim g n dµ n = g dµ. Then lim f n dµ n = f dµ. Lemma 51 (=Lemma 11.19). Every measurable subset of a positive set is positive. The union of countably many positive sets is positive. Lemma 52 (=Lemma 11.20). Let ν be a signed measure. If E is a set such that 0 < νe <, then there is a positive set A with A E and νa > 0. Theorem 53 (= Proposition 11.21, Hahn Decomposition Theorem). Let ν be a signed measure on a measurable space (X, B). Then there is a positive set A such that B = X A is a negative set. Lemma 54 (= Proposition 11.10, approximately). Let D be a countable dense subset of R, let (X, B, µ) be a measure space and suppose that, for every α D, there is B α B such that µ(b α B β ) = 0 for any α < β. Then there is a measurable function f such that f α on B α and f α on B α. If g is any other such function, then g = f a.e. Theorem 55 (=Theorem 11.23, Radon-Nikodym). Let (X, B, µ) be a σ-finite measure space and let ν be a measure on (X, B) which is absolutely continuous with respect to µ. Then there is a nonnegative measurable function f such that νe = f dµ for any E B. If g is another such function, then f = g a.e.[µ]. E

6 6 THEOREMS, ETC., FOR MATH 516 Proposition 56 (=Proposition 11.24, Lebesgue Decomposition Theorem). Let (X, B, µ) be a σ-finite measure space and let ν be a σ-finite measure on (X, B). Then there are two measures ν 0 and ν 1 such that ν = ν 1 + ν 2, ν 0 and µ are mutually singular, and ν 1 is absolutely continuous with respect to µ. These measures are unique. Lemma 57 (=Lemma 11.27). Let (X, B, µ) be a finite measure space, let g L 1 (µ), let p [1, ), and suppose that there is a constant M such that gϕ dµ M ϕ p for all simple functions ϕ. Then g L q (µ) for q = p/(p 1), and g q M. Theorem 58 (=Theorem 11.29, Riesz Representation). Let (X, B, µ) be a σ-finite measure space, let p [1, ), and let F be a bounded linear functional on L p (µ). Then there is a unique g L q (µ) such that F (f) = fg dµ for all f L p (µ). Theorem 59 (=Theorem 11.30). Let (X, B, µ) be a measure space, let p (1, ), and let F be a bounded linear functional on L p (µ). Then there is a unique g L q (µ) such that F (f) = fg dµ for all f L p (µ). Lemma 60 (=Lemma 12.2). Let µ be a measure on A, an algebra of sets. If A A and A i is a sequence of sets in A such that A i A i, then µa i µa i. Lemma 61 (=Corollary 12.3, Lemma 12.4, Lemma 12.5). Let µ be a measure on A, an algebra of sets. (a) If A A, then µ A = µa. (b) µ is an outer measure. (c) If A A, then A is µ -measurable. Proposition 62 (=Proposition 12.6). Let µ be a measure on A, an algebra of sets, and let E X. (i) For any ε > 0, there is A A σ such that E A and µ A µ E + ε. (ii) There is B A σδ such that E B and µ E = µ B. Proposition 63 (=Proposition 12.7). Let µ be a σ-measure on A, an algebra of sets. (i) A set E X is µ -measurable if and only if there are sets A A σδ and B with µ B = 0 such that E = A B. (ii) If µ B = 0, then there is a set C A σδ such that B C and µ C = 0. Theorem 64 (=Theorem 12.8, Carathéodory). Let µ be a σ-measure on A, an algebra of sets. Then µ, the restriction of µ to the collection of all µ -measurable sets, is a measure (on a σ-algebra of sets) which agrees with µ on A. If µ is finite (or σ-finite), then so is µ. If µ is σ-finite, then µ is the only such extension of µ on the σ-algebra generated by A.

7 THEOREMS, ETC., FOR MATH Proposition 65 (=Proposition 12.9). Let C be a semi-algebra and let µ be a nonnegative set function defined on C with µ = 0 if C. Suppose also that (i) If C C is the union of a finite collection {C 1,..., C n } of disjoints elements of C, then µc = µc i. (ii) If C C is the union of a countable collection {C 1,... } of disjoints elements of C, then µc µc i. Then µ has a unique extension to a measure on A, the algebra generated by C. Lemma 66 (=Lemma 12.14). Let {R i } be a countable collection of disjoint measurable rectangles with R = R i a measurable rectangle. Then λ(r) = λ(r i ). Lemma 67 (=Lemma 12.15). Let E R σδ and x X. Then E x is a measurable function on Y. Lemma 68 (=Lemma 12.16). Let E Rσδ with (µ ν)(e) <. Then g, defined by g(x) = ν(e x ), is a measurable function of x and g dµ = (µ ν)(e). Lemma 69 (=Lemma 12.17). If (µ ν)(e) = 0, then ν(e x ) = 0 for almost all x. Proposition 70 (=Proposition 12.18). Let E be a measurable subset of X Y with (µ ν(e) <. Then, for almost all x, E x is a measurable subset of Y and g, defined by g(x) = ν(e x ), is a measurable function of x. Moreover g dµ = (µ ν)(e). Theorem 71 (=Theorem 12.19, Fubini). Let (X, A, µ) and (Y, B, µ) be σ-finite measure spaces. If f is integrable with respect to µ ν, then (i) For almost all x X, the function f x, defined by f x (y) = f(x, y) is measurable, (i) For almost all y Y, the function f y, defined by f y (x) = f(x, y) is measurable, (ii) The function F 1, defined by F 1 (x) = f(x, y) dν is in L 1 (X), (ii) The function F 2, defined by F 2 (y) = is in L 1 (Y ), (iii) [ ] f dν dµ = [ X Y Y X ] f dµ dν = Y X f(x, y) dµ X Y f d(µ ν). Theorem 72 (=Theorem 12.20, Tonelli). Let (X, A, µ) and (Y, B, µ) be σ-finite measure spaces. If f is nonnegative and measurable with respect to µ ν, then (i) For almost all x X, the function f x, defined by f x (y) = f(x, y) is measurable, (i) For almost all y Y, the function f y, defined by f y (x) = f(x, y) is measurable,

8 8 THEOREMS, ETC., FOR MATH 516 (ii) The function F 1, defined by F 1 (x) = is in L 1 (X), (ii) The function F 2, defined by F 2 (y) = is in L 1 (Y ), (iii) [ ] f dν dµ = [ X Y Y X ] f dµ dν = Y X f(x, y) dν f(x, y) dµ X Y f d(µ ν). Proposition 73 (=Proposition 12.40). If µ is a Carathéodory outer measure with respect to Γ, then every function in Γ is µ -measurable. Proposition 74 (=Proposition 12.41). Let (X, ρ) be a metric space and let µ be an outer measure on X such that µ A + µ B = µ (A B) whenever ρ(a, B) > 0. Then every Borel set is measurable with respect to µ. Lemma 75 (=Lemma E5.2.1). If u has a weak derivative D α u for some multiindex α, then this weak derivative is unique. Theorem 76 (=Theorem E5.2.1). Suppose u and v are in W k,p (Ω) for some nonnegative integer k and some p [1, ]. (i) For any multi-index α with α k, we have D α u W k α,p (Ω) and D α (D β u) = D β (D α u) = D α+β u if α + β k. (ii) For any real numbers λ and µ, λu + µv W k,p and D α (λu + µv) = λd α u + µd β v. (iii) If V is a connected open subset of Ω, then u W k,p (V ). (iv) If ζ C c (Ω), then ζu W k,p (Ω) and D i (ζu) = D i ζu + ζd i u.