This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Modern Analysis, Homework 2 Due July 18, 2011 1. [20] (3 points) Rudin, Ch 7, #18: Let { f m } be a uniformly bounded sequence of functions which are Riemannintegrable on [ a,b ], and for each x [ a,b ], put F n ( x ) = Z x a f n ( t ) dt. Prove that a there exists a subsequence { F n k } which converges uniformly on [ a,b ]. 2. [21] (4 points) Rudin, Ch 7, #19: Let K be a compact metric space and S C ( K ). Prove that S is compact if and only if S is uniformly closed, pointwise bounded, and equicontinuous. ( Hint: If S is not equicontinuous, then S contains a sequence which has no equicontinuous subsequence, hence has no subsequence that converges uniformly on K .) 3. [27] (4 points) Rudin, Ch 8, #1: Define f ( x ) = e 1 /x 2 if x 6 = 0 , if x = 0 . Prove that f has derivatives of all orders at x = 0, and that f ( n ) (0) = 0 for all n N . ( Hint: You probably dont want to actually compute a formula for each f ( n ) ( x ). What form does each take?) 4. [32] Given reals4....
View
Full
Document
 Summer '11
 Staff

Click to edit the document details