This question just pop up when I was trying to solve another problem. Let $X,Y$ be vector topological spaces over the field $\mathbb{K}$ ( with $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ ) and let $f:X\to Y$ be linear. Now suppose $f$ is also an isomorphism in the algebraic sense.
I know this is not enough to guarantee that $f$ is also a homeomorphism. But what about the specific case $Y=\mathbb{K}$? It's clear that $f$ is continuous in this case, because for linear functionals we have that $$f\ \ \textrm{ is continuous} \iff Ker\ f \ \ \textrm{ is closed},$$
and this is the case here. Now we just need to show that $f^{-1}$ is also continuous and that is the strange part. I would like to say also that $f^{-1}:\mathbb{K}\to X$ is a linear functional and use the same result, but to do that I need to "see" $\mathbb{K}$ as topological vector space over $X$. This seems possible, for $X$ is algebraically isomorphic to $\mathbb{K}$, so $X$ is also a field.
This looks right but feels strange, could someone clarify this issue? I know the question looks vague, if you don't want to "talk about" a little about what could be making me strange this, I will be happy just in knowing if my reasoning is right.
Thanks.