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.