Selçuk Demir WS 2017 Functional Analysis Homework Sheet 1. Let M be a metric space. If A M is non-empty, we say that A is bounded iff diam(a) = sup{d(x, y) : x.y A} exists. Show that A is bounded iff there is a ball B(x, R) which contains A. Show also that A is bounded iff for every x M there is some R > 0 such that A B(x, R). 2. Show that any totally bounded set is bounded. 3. Suppose that X, Y are metric spaces and K > 0. If f : X Y and d(f(x), f(y)) K d(x, y) for every x, y M, we say that f is K-Lipschitz. Show that any Lipschitz function is continuous. 4. Let X be a metric space, p X be fixed. Define f : X R by f(x) = d(x, p) for every x X. Shoow that f is continuous. 5. Let M be a metric space, A be non-empty. Define f : M R by f(x) = d(a, x) = inf{d(x, a) : a A}. Show that f is continuous. 6. Let K, Y be metric spaces, f : K Y be a continuous bijection. Show that f 1 is also continuous. (We say that f is a homeomorphism). 7. Let K X be compact, p K c. Show that there is at least one x K such that d(p, x) = d(p, K). 8. Suppose that K is a compact metric space, f : K K is a continuous map. If f does not have any fixed point in K, show that there exists some ε > 0 such that d(x, f(x)) > ε for every x K. 9. Let A X and p X. Then p A iff d(p, A) = 0. 10. If A X is closed, show that there exists a continuous function f : X R such that A = f 1 ({0}). 11. If A, B R 2, define A + B = {a + b : a A and b B}. Show that if A or B is open, then A + B is open. If A and B are closed, show that A + B does not have to be closed. If A is closed and B is compact, show that A + B is closed. If A and B are compact, show that A + B must also be compact. 12. Let U OP(X). Show that, if A X, then A U = iff A U =. 1
13. Suppose that A, B X and that A B = A B =. Show that there exists some U, V OP(X) such that A U and B V with U V =. 14. Show that if U, V OP(X) are dense, then U V is also dense. 15. Let X be a metric space, A X. Prove that (A) c = (A c ) o. 16. Let A X and B Y be compact. Show that A B is compact in X Y. 17. Show that a compact metric space is separable. 18. Suppose that X is a metric space such that every closed ball is compact in X. Show that X is complete. Show also that X must be separable. 19. Suppose that X is a separable metric space and (U α ) is a family of pairwise disjoint family of open non-empty sets. Show that the family must be countable. 20. Let X = (C([0, 1]), d ). Define Λ(f) = 1 for every f X. Show that Λ is continuous. 0 f(x)dx 21. Let X be a metric space, K, E X be two non-empty disjoint closed subsets. Show that we can have d(k, E) = inf{d(k, e) : k K, e E} = 0. If K is also compact, then d(k, E) > 0. 22. Let X be a metric space, x, y X with x y. Show that there exists a continuous f : X [0, 1] such that f(x) = 0 and f(y) = 1. 23. Let X be a metric space, A X be a non-empty closed and p X A. Show that there exists a continuous f : X [0, 1] such that f(x) = 0 for every x A and f(p) = 1. 24. Let X be a metric space A, B X be two non-empty and disjoint closed subsets of X. Show that there exists a continuous function f : X [0, 1] such that f(x) = 0 for every x A and f(y) = 1 for every y B. 25. Let X be a metric space, A be a non-empty closed subset of X. Show that there exists a continuous function f : X R such that A = f 1 (0). 26. Let O(n, R) denote the set of orthogonal matrices in M(n, R). Show that O(n, R) is compact. 27. Let X be a complete metric space, f : X X be a function such that f n is a contraction for some n N. Show that f has a fixed point. 2
28. Let GL(n, R) denote the set of invertible matrices in M(n, R). Show that GL(n, R) is open. 29. Let X be a metric space and (x n ) be a sequence in X such that d(x n, x m ) > 1/2 whenever n m. Show that (x n ) cannot have any convergent subsequence. 30. Let X, Y be metric spaces, f, g : X Y be continuous functions. Define ϕ : X R by ϕ(x) = d(f(x), g(x)) for every x X. Show that ϕ is continuous. 31. Let X be NVS. A X is said to be convex if for every x, y A and λ (0, 1) we have λx + (1 λ)y A. Show that any ball is convex. 32. If A is convex, show that A is also convex. 33. Suppose that T : X Y is linear. Show that inf{m > 0 : T (x) M x x X} = sup T (x) = sup T (x). x 1 x <1 34. Let X, Y be Banach spaces. Suppose that T : X Y is such that there exists some α > 0 with T (x) α x for every x X. Show that T is 1-1 and that T (X) is a closed subspace of Y. 35. Let X be a normed vector space and Y X be a subspace. Show that Y is also a subspace. 36. Let X be a normed vector space, Y X be a subspace. Show that if Y contains a ball in X, then Y = X. 37. Consider the space C c (R). For f C c (R) define Λ(f) = f(x)dx. Show that Λ(f) is defined and defines a bounded linear functional on C c (R). 38. Let K = [a, b] [c, d] R 2. Let f C(K). If x [a, b], define ϕ f (x) = d f(x, t)dt. c Show that ϕ f C([a, b]). Deduce that I(f) := ( b ) d f(x, y)dy dx is defined. Show a c that I is a bounded linear functional. 39. Let X, Y be normed vector spaces, D X be a dense subspace of X. If T : D Y is a bounded linear operator, show that it has a unique extension to X which is also a bounded linear operator. 3
40. Let C c (R) denote the space of continuous functions f : R C such that f = 0 outside a compact subset of R. Show that (C c (R), ) is not complete. 41. Let C 0 (R) denote the space of continuous complex functions on R which satisfy the following property: ε > 0 M > 0 such that f(x) < ε for every x M. Show that f is defined and C c (R) is a Banach space with respect to the sup-norm. 42. Let X be a normed vector space. If A, B X, define A + B to be the set of all elements of the form x + y where x A and y B. If A is open, show that A + B is open. 43. If A is compact and B is closed, show that A + B is closed. 44. Give an example of a normed vector space X and two closed A, B X such that A + B is not closed. 45. Let X and Y be normed vector spaces and T : X Y be linear. Show that T is continuous iff T is bounded on some ball B in X. 46. Let X be the space of all real polynomials in one variable. If P (x) = n i=0 a ix i X, show that P 1 = i a i and P = max i a i define two norms on X which are not equivalent. 47. Let T : X Y be a continuous linear operator. Show that it graph, Γ(T ) := {(x, T (x)) : x X}, is a closed subspace of X Y. 48. Show that (x n, y n ) converges to (x, y) iff x n x and y n y. 49. Show that A B X Y is compact iff A X and B Y are compact. 50. Show that X Y is separable if both X and Y are separable. 51. Show that X Y is complete if both X and Y are complete. 52. Show that any finite dimensional vector subspace of a normed vector space is closed. 53. Let f C([0, 1]). Define ( 1 f p = 0 ) 1/p f(x) p dx for 1 p <. Show that this defines a norm on C([0, 1]). 54. Show that C([0, 1]) with the norm 1 is not complete. 55. Let X and Y be normed vector spaces. Show that a linear T : X Y is continuous iff it maps Cauchy sequences to Cauchy sequences. 4
56. Let X be an infinite-dimensional normed vector space. Show that there exists a linear functional f : X K which is not bounded. 57. Let V be a finite-dimensional Banach space, W a normed vector space. Show that every linear T : V W is bounded. Deduce that a normed vector space X is finite-dimensional iff every linear functional on X is bounded. 58. Suppose that X is a normed vector space and D is a countable subset of X such that the vector subspace D generated by D is dense in X. Show that X is separable. 59. Is the Banach space C([0, 1]) separable? 60. Is the Banach space l p separable? 61. Is the Banach space c 0 separable? 62. If A N, define x A l be the characteristic function of A. Show that {B(x A, 1/2) : A N} is an uncountable family of pairwise disjoint, non-empty open balls in l. Deduce that l is not separable. 63. Show that any finite-dimensional normed vector space is a Banach space. 64. Let K be a compact metric space. Is C(K) separable? 65. Suppose that X and Y are Banach spaces, U OP(X). f : U Y is said to be differentiable at p U if there exists a bounded linear T : X Y such that f(p + h) f(p) T (h) lim h 0 h Show that T is unique if f is differentiable at p. 66. If f in the previous problem is differentiable at p, show that it is continuous at p. = 0. 67. Suppose that (X n ) is a sequence of metric spaces, X = n N X n, and d(x, y) = n N 1 2 n d n (x n, y n ) 1 + d n (x n, y n ) for every x = (x n ) and y = (y n ). Let (x k ) be a sequence in X. Show that x k x in X iff x k n x n for every n N. Show also that the projection mappings p n : X X n given by p n (x) = x n is continuous and open. Show also that X is compact iff each X n is compact. 5