HW2 - Mathematics Department Stanford University Math 175...

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Mathematics Department Stanford University Math 175 Homework 2 1. Suppose X is a Banach space (i.e. a complete normed linear space) and S is a linear subspace of X . Prove that S , viewed as a normed linear space with the norm of X , is complete if and only if S , viewed as a subset of X , is closed. 2. Let X be a normed linear space and S a linear subspace with S ¤ X . Prove that S contains no ball of X . 3. Let M ¤ be a metric space (not necessarily a linear space) with metric d . (i) Given an open ball B ± .x/ , an open dense subset U of M , prove there is a ball B ² .y/ ± B ± .x/ \ U . (Note: Recall U dense means U D M .) (ii) If U 1 ;U 2 ;::: are open dense sets in M , prove there exist a “nested” sequence of open balls B ± 1 .x 1 / ² B ± 2 .x 2 / ² ³³³ such that ± j < 1 =j and B 2 ± j .x j / ± U j for each j D 1 ; 2 ;::: . Hint: (i) already provides the inductive step for this construction. 4. Let M ¤ be a complete metric space. Prove (i) If U 1 ;U 2 ;::: are open and dense, then \ 1 j D 1 U
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This note was uploaded on 02/01/2011 for the course MATH 171 taught by Professor R during the Fall '09 term at Stanford.

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