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

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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 0 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 0 is separable, where c 0 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 x 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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