Is the Frechet space of all real sequences locally compact? Is a Hilbert cube, viewed as a topological metric space locally compact?
Are there any infinitely dimensional locally compact spaces?
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general-topology
analysis
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5The Hilbert cube is compact hence locally compact. – 2012-09-22
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5every locally convex space which is locally compact has finite dimension. – 2012-09-22
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4So: 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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2Another 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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5What do you mean by infinite-dimensional here? – 2012-09-22