Question 2. 25 marks] (Problem 38 in Section 13.4 of Royden and Fitzpatrick's book) Let (Y, ll-ll...
Question 2. 25 marks] (Problem 38 in Section 13.4 of Royden and Fitzpatrick's book) Let (Y, ll-lly) be a normed vector space. Show that (Ý, IHIY) is Banach if and only if there exist a Banach space (X, l-1x) and an open operator T E (X,Y) Hint: You may use without proof the following two facts (which we used in our proof of Riesz-Fischer's Theorem for LP-spaces): There erists a subsequence (vn)nN such that lv +) - vpt)ly S 2-" Vn EN. Up(n))nEN converges, then (vn)neN converges Furthermore, it may useful to write υφ(n) -Σ (w+1)-W)) + νφ(1).
Question 2. 25 marks] (Problem 38 in Section 13.4 of Royden and Fitzpatrick's book) Let (Y, ll-lly) be a normed vector space. Show that (Ý, IHIY) is Banach if and only if there exist a Banach space (X, l-1x) and an open operator T E (X,Y) Hint: You may use without proof the following two facts (which we used in our proof of Riesz-Fischer's Theorem for LP-spaces): There erists a subsequence (vn)nN such that lv +) - vpt)ly S 2-" Vn EN. Up(n))nEN converges, then (vn)neN converges Furthermore, it may useful to write υφ(n) -Σ (w+1)-W)) + νφ(1).