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Any real (Hausdorff) topological real vector space of dimension $n<\infty$ is isomorphic to $\mathbf{R}^n$ with the standard topology, see e.g. Rudin, Functional analysis, theorem 1.21.

Here are some comments:

  1. For some reason it is stated there for complex vector spaces, but, as remarked after the theorem, the proof works for real vector spaces as well.

  2. Instead of the Hausdorff axiom Rudin uses the (weaker) $T_1$ axiom in the definition of a topological vector space.
show/hide this revision's text 1

Any real (Hausdorff) topological real vector space of dimension $n<\infty$ is isomorphic to $\mathbf{R}^n$ with the standard topology, see e.g. Rudin, Functional analysis, theorem 1.21.

Here are some comments:

  1. For some reason it is stated there for complex vector spaces, but, as remarked after the theorem, the proof works for real vector spaces as well.

  2. Instead of the Hausdorff axiom Rudin uses the (weaker) $T_1$ axiom in the definition of a topological vector space.