Functional analysis Questions

81 17:35 Expert Q&A W + 3. Prove that the metric space ( of all bounded scalar sequences is complete. Recall that the metric on 2 is given by doo (x, y) = sup | - Vil, for all a = (6:), y = (7) Ex. iEN 4. The set co of all scalar sequences that converge to 0 is a subset of 190, since convergent sequences are bounded. Use problem 3 above together with Theorem 1.36 in the notes to show that co is a complete metric space (if we use the metric doo on co). 1. Let (X, d) and (Y, p) be metric spaces. Prove that a mapping f : X - Y is continuous if and only if the inverse images f ( G) := { EX : f(2) EG} of all closed sets G in Y are closed in X. 2. Refer to Definition 1.53 (of uniform continuity) in the lecture notes. Show that if we define f : (0, 1) - R by f(2) = 4, then f is continuous on the metric space ((0, 1), | . () but not uniformly continuous. 3. Let (X, d) be a metric space. We call two Cauchy sequences (In), (Un) in X equivalent if lim d(xn, Un) = 0. 7-+00 In this case we write (an) ~ (Un). Prove that ~ is an equivalence relation, i.e. prove that (a) (In) ~ (In). ( b ) ( in) ~ ( yn) = (Un) ~ (In). ( c ) ( ( In ) ~ ( Un) & ( yn) ~ ( En) ) = (In) ~ (27) 4. Let (X, d) be a metric space and (In) a Cauchy sequence in X. De- note by a the set (or family) of all Cauchy sequences in X which are equivalent to (an) (see Problem 3 above), i.e. ( Un) Ex - (yn) ~ (In). We call a an equivalence class of Cauchy sequences in X. Let X denote the set of all equivalence classes of Cauchy sequences in X. (i) Define d : X x X - R by d(x, y ) := lim d(In, yn), n -+ 00 where (In) Ex and (Un) Ey. Show that for every choice of (In) E x and (Un) E y, we always get the same limit lim d(in, yn), i.e. that the limit is independent of the choice of the Cauchy sequences. This will then imply that d is well defined. (ii) Then show that (X, d) is a metric space. O3. Prove that the metric space ( of all bounded scalar sequences is complete. Recall that the metric on 2 is given by doo (x, y) = sup | - vil, for all a = (6:), y = (71) ex. iEN 4. The set co of all scalar sequences that converge to 0 is a subset of 20, since convergent sequences are bounded. Use problem 3 above together with Theorem 1.36 in the notes to show that co is a complete metric space (if we use the metric doo on co). 1. Let (X, d) and (Y, p) be metric spaces. Prove that a mapping f : X - Y is continuous if and only if the inverse images f ' (G) := { EX : f(z) EG} of all closed sets G in Y are closed in X. 2. Refer to Definition 1.53 (of uniform continuity) in the lecture notes. Show that if we define f : (0, 1) - R by f(2) = 4, then f is continuous on the metric space ((0, 1), | . () but not uniformly continuous. 3. Let (X, d) be a metric space. We call two Cauchy sequences (In), (Un) in X equivalent if lim d(an, Un) = 0. 1-+00 In this case we write (an) ~ (Un). Prove that ~ is an equivalence relation, i.e. prove that (a) (In) ~ (an). ( b ) ( an) ~ (Un) = (Un) ~ (In). ( c ) ( ( In ) ~ ( Un) & ( Un) ~ (En) ) = (In) ~ (27)