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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?

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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
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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

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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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Btw, this is the second time I see it: can one really say "locally convexity"? It makes me twitch every time I hear it. What about "local convexity"? Is that also acceptable? Or is locally convexity the correct term? –  Rudy the Reindeer Aug 10 '12 at 10:10
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Local convexity is definitely not irrelevant... You can drop it but at the heavy cost of metrizability which is rather a strong requirement, but you can also get away with a variant called (ultra)barrelledness + a little more. @Matt: It is "local convexity". –  t.b. Aug 10 '12 at 10:13
    
@t.b. Ah, thanks. Then I'll keep on twitching... (assuming I'll be seeing it again) : ) –  Rudy the Reindeer Aug 10 '12 at 10:14
    
@Matt Yes you are right, it sounds awefull. –  Alexander Thumm Aug 10 '12 at 10:15
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No, 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. –  t.b. Aug 10 '12 at 10:25

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