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?
Is inverse mapping theorem true for locally convex spaces
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functional-analysis
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1The 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. – 2012-08-10
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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.
Local convexity is also quite irrelevant to the problem. The theorem is generally true for F-spaces, that is complete metric vector spaces, where the metric is compatible with the vector space operations. Locally convex F-spaces are Fréchet spaces.
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.
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2No, no, never mind. The answer is fine for such a minimalistic question (you got +1 from me). I was just trying to make the point that local convexity is rather useful since it simplifies matters quite a bit. A very (almost painfully) detailed account of the necessary and sufficient conditions for the standard Baire-consequences to hold is given in Schechter's handbook, see theorems 27.26 and 27.27 on page 734, [Google books link here](http://books.google.com/books?id=eqUv3Bcd56EC&pg=PA734). – 2012-08-10