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Is the Frechet space of all real sequences locally compact? Is a Hilbert cube, viewed as a topological metric space locally compact?

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    The Hilbert cube is compact hence locally compact.2012-09-22
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    every locally convex space which is locally compact has finite dimension.2012-09-22
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    So: note that the Hilbert cube (inside $\mathbb R^\infty$) is not a neighborhood, so (even though the Hilbert cube is compact) this does not suggest $\mathbb R^\infty$ is locally compact.2012-09-22
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    Another example: the disjoint union $\coprod_{n\in\mathbb N} S^n$ of $n$-dimensional spheres is locally compact and has infinite dimension.2012-09-22
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    What do you mean by infinite-dimensional here?2012-09-22

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