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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....
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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.
- Spring '10