hmwk2 - MATH 202A HOMEWORK 2 This assignment covers...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 202A HOMEWORK 2 This assignment covers material from the second week of class. The corresponding reading material is Royden Ch 7, 3-7. Problem 1 . Show that separability is a topological property, i.e. that it is preserved under homeomorphism. In the next problem, the separability of various classical spaces is examined. Recall that in class, we proved that ` is not separable. Note the contrast with (c) below, where you show that a special subspace c of ` is separable. In (d), you show that ` 2 is separable, but it is in fact more generally true that ` p is separable for 1 p < . Problem 2 . (a) Show, using the Weierstrass theorem without proof, that C ([ a, b ]) is separable. (b) Is C b ( R ) separable? Prove or disprove. Here, C b ( R ) is the space of all bounded continuous C-valued functions defined on R with the supremum norm. (c) Show that c is separable, where c is the subspace of ` consisting of all x = ( x m ) ` such that lim m + x m = 0 (d) Show that ` 2 is separable. Here, ` 2 is the space of (infinite) sequences x = ( x m ) with the norm k x k ` 2 = ( + m =1 | x m | 2 ) 1 / 2 . Let us recall some definitions. C b ( R ) is the space of bounded continuous functions on R with the supremum norm, i.e....
View Full Document

This note was uploaded on 04/06/2010 for the course MATH various taught by Professor Tao/analysis during the Spring '10 term at UCLA.

Page1 / 3

hmwk2 - MATH 202A HOMEWORK 2 This assignment covers...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online