Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
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

1 Answer 1

up vote 3 down vote accepted

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.

share|improve this answer
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
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
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.