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?
As pointed out by Matt in the comments, the bounded inverse theorem makes use of completeness of both the domain and codomain and t.b. gives a counterexample there.
Update: t.b. has made a point in the comments, that local convexity is not irrelevant since under this hypothesis, one can drop the metrizability requirement and replace is with something weaker.