Mich 2017 FUNCTIONAL ANALYSIS - EXAMPLES 2 AZ 1. Let (2, F, p) be a o-finite measure space, and let v: F -> C be a complex measure. Show that there exist unique complex measures v1 and v2 on F such that v = V1 + V2, V1 « ? (v1 absolutely continuous with respect to ?) and V21? (v2 orthogonal to ?). Recall that u and v2 are said to be orthogonal if there is a measurable partition 12 = AUB such that u(B) = [/2|(A) = 0.] 2. Let K be a compact Hausdorff space, and (fn) a bounded sequence in C(K). Show that (fn) is weakly null if and only if it is pointwise null. 3. Show that there is a unique regular Borel measure À on R such that A ([a, b]) = b - a for all a < b in R. 4. Let X be a topological space with topology T. Let F be a family of functions such that each f E F is a function (not necessarily continuous) from X to some topological space Yf. Assume that for every topological space Z a function g : Z -> X is continuous if and only if f o g is continuous for all f E F. Show that T = 0(X, F). 5. Show that a Banach space X with its weak topology is a completely regular topological space: given a weakly closed subset C of X and p & C, there is a continuous function f: (X, w) -> [0, 1] such that f (p) = 1 and f = 0 on C. 6. Show the following quantitative version of Lemma 3.3. Let X be a normed space, f, g1, 92, ... , gn E X0 and a > 0. Assume that the restriction of f to ("i=1 ker gi has norm at most E. Deduce that d (f, span{g1, ... , gn} ) 6 E. 7. Show that the weak- and norm-topologies coincide iff dim X < oo. Show that the weak topology of an infinite-dimensional normed space and the w"-topology of the dual space of an infinite-dimensional Banach space are not metrizable. [Hint: assume they are, and consider open balls at 0 with radius 1/n, n E N.] 8. Is the canonical embedding XI -> X" w"-to-w" continuous? 9. Let T: X -> Y be a linear map between normed spaces. Show that TFAE (i) T is norm-to-norm continuous. (ii) T is weak-to-weak continuous. (iii) T is norm-to-weak continuous. Show further that T is weak-to-norm continuous if and only if T is bounded and of finite-rank. 10. Let T: YI -> X" be a bounded linear map between dual spaces. Show that TFAE (i) T is w"-to-w" continuous. (ii) There is a bounded linear map S: X -> Y with SI = T. (iii) T'(X) CY. 11. Prove that the closed unit ball Bx of a normed space X is weakly closed. Use this to show that if every norm-Cauchy