Math 108a 2011 Assignment 6 1. (a) It is enough to con.rm claim for positive decreasing sequence of functions (fn )1 n=0 with fn (x) fn+1 (x) for all x 2 X which goes to 0. Let " > 0 and consider open sets Un = fx 2 X : fn (x) < "g. Collection fUn : n 0g
MAT 108A-HW7 November 21. 2011. Due date: Tuesday November 29. 2011. at 4pm 1. (20)(Principle of uniform bound) Let X be a Banach space, and let Y be a normed vector space. Let F B(X; Y ). Show that if for every x 2 X we have supA2F jjAxjj < 1 then supA2F
MAT 108A-HW5 November 8. 2011. Due date: Tuesday November 15. 2011. at 4pm 1. (20)Let V be an in.nite dimensional normed vector space, and suppose that V = Wn is a .nite dimensional vector subspace of V . Prove that V is not complete.
1 S
Wn , where each
MAT 108A-HW4 November 1. 2011. Due date: Tuesday November 8. 2011. at 4pm 1. (20)Let (X; d) be a complete metric space and let G X be an open subset. Show that there is a complete metric on G such that it has the same open sets as G has with relative metr
Math 108a 2011 Assignment 4 1. Without loss of generality we may assume that closed set F = XnG is nonempty. We de.ne metric on G with (x; y) = d(x; y) + j 1 d(x; F ) 1 j. d(y; F )
It is straightforward to check that this is a metric. Since d(x; y) (x; y)
MAT 108A-HW6 November 15. 2011. Due date: Tuesday November 22. 2011. at 4pm 1. (20) (a) Let (X; d) be a compact metric space, and suppose that (fn )1 is a pointwise increasing sequence n=0 in C(X), i.e. fn (x) fn+1 (x) for all x 2 X and all n 0. If pointw
Math 108a 2011 Assignment 7 1. Let Bn = fx 2 X : jjAxjj n for all A 2 Fg. Since supA2F jjAxjj < 1 for all x 2 X, we have X = [1 Bn . Each Bn is closed as intersection of closed sets Bn = \A2F fx 2 n=1 X : jjAxjj ng. Since every Banach space is of the seco
Math 108a 2011 Assignment 5 1. Without loss of generality we may assume that W1 W2 :.Let d be a metric induced by norm. We may take w1 2 W1 such that w1 6= 0. Then there is an w2 2 W2 nW1 1 . Then there is an w3 2 W3 nW2 such that such that d(w1 ; w2 ) <