I want to show the following: Let $V$ be a finite dimensional topological vector space and Hausdroff. Suppose $f: \mathbb{K}^n \rightarrow V$ to be an vector space isomorphism. Show that $f$ is a homeomorphism.
Since $f$ is a vector space isomorphism, we know that it is bijective. So the only thing left to show is that $f$ and $f^{-1}$ is continuous. Further, since $f$ is an isomorphism between $V$ and $\mathbb{K}^n$, we can conclude that $dim V=n$.
Let $B \subseteq V$ be an open subset. I want to show that $f^{-1}(B)$ is open in $\mathbb{K}^n$. What do I also know: The addition $+:V \times V \rightarrow V$ and scalar multiplication $\cdot:\mathbb{K} \times V \rightarrow V$ are continuous.
I do not really know how to continue from here on. Hints/Solution would be appreciated.
I am not sure if it is correct, but here are my thoughts:
Let $\{b_1,...,b_n\}$ be a basis of $\mathbb{K}^n$, a vector space isomorphism is determined by how it maps $\{b_1,...,b_n\}$ to $V$. Let $x \in \mathbb{K}^n$, i.e $x=\sum_{k=1}^{n}\lambda_k b_k$. Then $T(x)=\sum_{k=1}^{n}\lambda_k T(b_k)$ and since the addition in $V$ and scalar multiplication is continuous (V is a topological vector space) we can conclude that $T$ is continuous.
A vector space isomorphism is a linear bijection, thus can be inverted. By the same argumentation, $T^{-1} : V \rightarrow \mathbb{K}^n$ is continuous.