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There are two versions of second category space: one is complete metric space, the other is locally compact space.

As we know, an open interval is locally compact but not complete. But how about the opposite? Must a complete space be locally compact? If not, must a complete topological vector space be second category?

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    A Banach space is locally compact iff it is finite-dimensional.2012-04-21
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    For the first question consider any infinite dimensional Banach space, for the second I think $C_c(\mathbb{R})$ (continous real functions with compact support) with the limit topology is a counterexample.2012-04-21
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    @martini May I suggest you post your answer as an answer and not as a comment?2012-04-22
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    @Jose27 May I suggest you post your answer as an answer and not as a comment?2012-04-22
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    @MartinArgerami done so.2012-04-23
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    @martini: thanks!2012-04-23

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