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my question is related to the very first part of Chapter 3 of Resnicks book "Extreme Values, Regular Variation and Point Processes. He is claiming that one should think of a locally compact Hausdorff space with countable base as a subset of a compactified finite dimensional euclidean space (which is a topological vector space).

Indeed, if we were in a locally compact Hausdorff topological vector space with countable base this would be an obvious statement since every locally compact topological vector space is finite dimensional. My question is: if we additionally assume that the locally compact Hausdorff space with countable base is a vector space do we automatically get that this space is also a topological vector space?

The mathematical formulation of the problem: Let $X$ be a vector space over $K$. Assume that $X$ is Hausdorff, locally compact and has a countable base. Is $X$ a topological vector space?

I tried to tackle the problem using that $X$ is metrizable and separable, but this did not lead me anywhere.

Thanks!

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  • $\begingroup$ @HennoBrandsma I am a little confused. Why is multiplication not continuous in this case? I assume that $\mathbb{R}$ and $\mathbb{Q}$ are endowed with the standard topology. $\endgroup$ – Florian Brück Dec 11 '19 at 8:48
  • $\begingroup$ I was confused. The finite dimensionality for locally compact TVS’s only holds for realms and complex scalars. $\endgroup$ – Henno Brandsma Dec 11 '19 at 9:22
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This is obviously false, because the topology could have nothing to do with the vector space structure. For instance, with $K=\mathbb{R}$ and $X=\mathbb{R}$ with its usual vector space structure, we can pick a bijection between $X$ and $[0,1]$ and thus give $X$ a topology which is homeomorphic to the usual topology on $[0,1]$. Or, we could pick some crazy bijection between $X$ and $\mathbb{R}$ to put a topology on $X$ which is homeomorphic to the usual topology but for which addition is not continuous.

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  • $\begingroup$ Thanks! That makes sense. $\endgroup$ – Florian Brück Dec 11 '19 at 8:46

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