# Is inverse mapping theorem true for locally convex spaces

If $X,Y$ are locally convex spaces, and $f:X\rightarrow Y$ is a continuous linear transformation which is bijective, then is the inverse of $f$ continuous as well?

-
Of course not. Let $X$ be normed and infinite-dimensional and $Y = X$ be equipped with the weak topology $\sigma(X,X^\ast)$ which is strictly weaker than the norm topology. –  t.b. Aug 10 '12 at 9:52
The property you exploit in the proof of the open mapping theorem is completeness of domain and codomain. So perhaps if you take $X,Y$ to be Fréchet spaces (or perhaps other complete TVSs), the answer might be yes. Unfortunately, I don't know enough about the topic to answer that question. –  Rudy the Reindeer Aug 10 '12 at 9:54