Institut für Analysis WS2019/20 Prof. Dr. Dorothee Frey 24.10.2019 M.Sc. Bas Nieraeth Functional Analysis Exercise Sheet 2 Remark. The exercises marked with a ∗ can be handed in for correction at the beginning of the exercise class next week. ∗ Exercise 1: Relative metrics Let (X; d) be a metric space and let M ⊆ X be equipped with the relative metric dM , i.e., the restriction of d to M × M. (a) Show that A ⊆ M is open in M if and only if there exists a set A0 ⊂ X that is open in X satisfying A = A0 \ M. Prove an analogous result for closed sets. M (b) For a subset A ⊆ M we denote the closure of A with respect to dM as A and the closure M with respect to d as A. Show that A = A \ M. (c) (Tricky) Suppose that X is separable. Prove that M is also separable. ∗ Exercise 2: Separability in normed spaces Let (X; k · k) be a normed space. Show that the following are equivalent: (i) X is separable; (ii) The unit ball BX := fx 2 X : kxk < 1g is separable; (iii) The unit sphere SX := fx 2 X : kxk = 1g is separable. Exercise 3: Separability and density under continuous maps Let (X; d) and (Y; ρ) be metric spaces and let f : X ! Y be continuous. (a) Suppose that A ⊆ X is dense in X. Show that f(A) is dense in f(X). (b) Show that if X is separable, then so is f(X). Exercise 4: Failure of completeness of R with an equivalent metric Consider the real line R equipped with the metric d(x; y) := j arctan(x) − arctan(y)j. (a) Let (xn)n2N ⊆ R be a sequence and x 2 R. Show that limn!1 d(xn; x) = 0 if and only if limn!1 jxn − xj = 0, i.e., show that d is equivalent to the usual metric on R. (b) Show that (R; d) is not complete. — Turn the page! — ∗ Exercise 5: Point evaluations on C([0; 1]) Consider the space C([0; 1]) equipped with the supremum norm k · k1. (a) Let x 2 [0; 1] and define the evaluation map evx : C([0; 1]) ! R by evx(f) := f(x). Prove that evx is continuous. (b) Using the evaluation maps, prove that the following sets are closed in C([0; 1]): (i) A = ff 2 C([0; 1]) : f(x) 2 F g, where x 2 [0; 1] and F ⊆ R is a closed set. (ii) B = ff 2 C([0; 1]) : f(x) ≥ 0 for all x 2 [0; 1]g. http://www.math.kit.edu/iana3/edu/fa2019w/de.