Functional analysis 1 Winter semester 2013-14 1. Topological vector spaces Basic notions. Notation. (a) The symbol F stands for the set of all reals or for the set of all complex numbers. (b) Let (X; τ) be a topological space and x 2 X. An open set G containing x is called neigh- borhood of x. We denote τ(x) = fG 2 τ; x 2 Gg. Definition. Suppose that τ is a topology on a vector space X over F such that • (X; τ) is T1, i.e., fxg is a closed set for every x 2 X, and • the vector space operations are continuous with respect to τ, i.e., +: X × X ! X and ·: F × X ! X are continuous. Under these conditions, τ is said to be a vector topology on X and (X; +; ·; τ) is a topological vector space (TVS). Remark. Let X be a TVS. (a) For every a 2 X the mapping x 7! x + a is a homeomorphism of X onto X. (b) For every λ 2 F n f0g the mapping x 7! λx is a homeomorphism of X onto X. Definition. Let X be a vector space over F. We say that A ⊂ X is • balanced if for every α 2 F, jαj ≤ 1, we have αA ⊂ A, • absorbing if for every x 2 X there exists t 2 R; t > 0; such that x 2 tA, • symmetric if A = −A. Definition. Let X be a TVS and A ⊂ X. We say that A is bounded if for every V 2 τ(0) there exists s > 0 such that for every t > s we have A ⊂ tV . Definition. We say that a TVS space X is • locally convex if there exists a basis of 0 whose members are convex, • locally bounded if 0 has a bounded neighborhood, • metrizable if its topology is compatible with some metric on X, • F-space if its topology is induced by a complete invariant metric, • Fr´echet space if X is a locally convex F-space, • normable if a norm exists on X such that the metric induced by the norm is compatible with the topology on X. Theorem 1.1. Let (X; τ) be a TVS. (a) If K ⊂ X is compact, C ⊂ X is closed, and K \ C = ;, then there exists V 2 τ(0) such that (K + V ) \ (C + V ) = ;. (b) For every neighborhood U 2 τ(0) there exists V 2 τ(0) such that V ⊂ U. (c) The space X is a Hausdorff space, i.e., for every x1; x2 2 X; x1 6= x2; there exist disjoint open sets G1;G2 such that xi 2 Gi; i = 1; 2. Theorem 1.2. Let X be a TVS, A ⊂ X, and B ⊂ X. Then we have (a) A = TfA + V ; V 2 τ(0)g, (b) A + B ⊂ A + B, (c) if V is a vector subspace of X, then V is a vector subspace of X, (d) if A is convex, then A and int A are convex, (e) if A is balanced, then A is balanced; if moreover 0 2 int A, then int A is balanced, (f) if A is bounded, then A is bounded. 1 2 Theorem 1.3. Let X be a TVS. (a) For every U 2 τ(0) there exists balanced V 2 τ(0) with V ⊂ U. (b) For every convex U 2 τ(0) there exists balanced convex V 2 τ(0) with V ⊂ U. Corollary 1.4. Let X be a TVS. (a) The space X has a balanced local base. (b) If X is locally convex, then it has a balanced convex local base. Theorem 1.5. Let (X; τ) be a TVS and V 2 τ(0). S1 (a) If 0 < r1 < r2 < : : : and lim rn = 1, then X = n=1 rnV . (b) Every compact subset K ⊂ X is bounded. (c) If δ1 > δ2 > δ3 > : : : , lim δn = 0, and V is bounded, then the collection fδnV ; n 2 Ng is a local base for X. Linear mappings. Theorem 1.6. Let (X; τ) and (Y; σ) be TVS and T : X ! Y be a linear mapping. Then the following are equivalent. (i) T is continuous. (ii) T is continuous at 0. (iii) T is uniformly continuous, i.e., for every U 2 σ(0) there exists V 2 σ(0) such that for every x1; x2 2 X with x1 − x2 2 V we have T (x1) − T (x2) 2 U. Theorem 1.7. Let T : X ! F be a nonzero linear mapping. Then the following are equivalent. (i) T is continuous. (ii) ker T is closed. (iii) ker T 6= X. (iv) T is bounded on some V 2 τ(0). Metrization. Theorem 1.8. Let X be a TVS with a countable local base. Then there is a metric d on X such that (a) d is compatible with the topology of X, (b) the open balls centered at 0 are balanced, (c) d is invariant. If, in addition, X is locally convex, then d can be chosen so as to satisfy (a), (b), (c), and also (d) all open balls are convex. Corollary 1.9. Let X be a TVS. Then the following are equivalent. (i) X is metrizable. (ii) X is metrizable by an invariant metric. (iii) X has a countable local base. Theorem 1.10. (a) If d is an invariant metric on a vector space X then d(nx; 0) ≤ nd(x; 0) for every x 2 X and n 2 N. (b) If fxng is a sequence in a metrizable topological vector space X and if lim xn = 0, then there are positive scalars γn such that lim γn = 1 and lim γnxn = 0. Boundedness and continuity. Theorem 1.11. The following two properties of a set E in a topological vector space are equivalent: (a) E is bounded. (b) If fxng is a sequence in E and fαng is a sequence of scalars such that lim αn = 0, then lim αnxn = 0. 3 Theorem 1.12. Let X and Y be TVS and T : X ! Y be a linear mapping. Consider the following properties. (i) T is continuous. (ii) T is bounded, i.e., T (A) is bounded whenever A ⊂ X is bounded. (iii) If fxng converges to 0 in X, then fT (xn); n 2 Ng is bounded. (iv) If fxng converges to 0 in X, then fT (xn)g converges to 0. Then we have (i) ) (ii) ) (iii). If X is metrizable then the properties (i){(iv) are equivalent. Pseudonorms and local convexity. Definition. (a) A pseudonorm on a vector space X is a real-valued function p on X such that •8 x; y 2 X : p(x + y) ≤ p(x) + p(y)(subadditivity), •8 α 2 F 8x 2 X : p(αx) = jαjp(x). (b) A family P of pseudonorms on X is said to be separating if to each x 6= 0 corresponds at least one p 2 P with p(x) 6= 0. (c) Let A ⊂ X be an absorbing set. The Minkowski functional µA of A is defined by −1 µA(x) = infft > 0; t x 2 Ag: Theorem 1.13. Suppose p is a pseudonorm on a vector space X. Then (a) p(0) = 0, (b) 8x; y 2 X : jp(x) − p(y)j ≤ p(x − y), (c) 8x 2 X : p(x) ≥ 0, (d) fx 2 X; p(x) = 0g is a subspace, (e) the set B = fx 2 X; p(x) < 1g is convex, balanced, absorbing, and p = µB. Theorem 1.14. Let X be a vector space and A ⊂ X be a convex absorbing set. Then (a) 8x; y 2 X : µA(x + y) ≤ µA(x) + µA(y), (b) 8t ≥ 0: µA(tx) = tµA(x), (c) µA is a pseudonorm if A is balanced, (d) if B = fx 2 X; µA(x) < 1g and C = fx 2 X; µA(x) ≤ 1g, then B ⊂ A ⊂ C and µA = µB = µC . Theorem 1.15. Suppose B is a convex balanced local base in a topological vector space X. Asso- ciate to every V 2 B its Minkowski functional µV . Then fµV ; V 2 Bg is a separating family of continuous pseudonorms on X. Theorem 1.16. Suppose that P is a separating family of pseudonorms on a vector space X. Associate to each p 2 P and to each n 2 N the set 1 V (p; n) = x 2 X; p(x) < n : Let B be the collection of all finite intersection of the sets V (p; n). Then B is a convex balanced local base for a topology τ on X, which turns X into a locally convex space such that (a) every p 2 P is continuous, and (b) a set E ⊂ X is bounded if and only if every p 2 P is bounded on E. Theorem 1.17. Let X be a locally convex space with countable local base. Then X is metrizable by an invariant metric. Theorem 1.18. A TVS space X is normable if and only if its origin has a convex bounded neighborhood. 4 The Hahn-Banach theorems. Theorem 1.19. Suppose that A and B are disjoint, nonempty convex sets in a topological vector space X. (a) If A is open there exist Λ 2 X∗ and γ 2 R such that Re Λ(x) < γ ≤ Re Λ(y) for every x 2 A and for every y 2 B. ∗ (b) If A is compact, B is closed, and X is locally convex, then there exist Λ 2 X , γ1; γ2 2 R, such that Re Λ(x) < γ1 < γ2 ≤ Re Λ(y) for every x 2 A and for every y 2 B.