bio  website  cmcc.ufabc.edu.br/… 

location  Cotia, Brazil  
age  35  
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I'm a professor at the Center of Mathematics, Computation and Cognition of the Federal University of the ABC, Santo André, Brazil.
2d

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Is every Montel locally convex vector space compactly generated?
Oops, that was a typo (which unfortunately I can no longer edit out), sorry... Thanks! 
Oct 20 
comment 
Is every Montel locally convex vector space compactly generated?
I do not have the book of Frölicher and Kriegl at hand, but I guess these results are the ones you quoted, right? 
Oct 20 
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Is every Montel locally convex vector space compactly generated?
Hmm... Come to think of it, this actually could be inferred indirectly from the discussion in the book of KrieglMichor (cited in the question) as well: Theorem 4.11 (3), pp. 3940 states that the Kelley topology of a strict inductive limit of a sequence of Fréchet spaces coincides with its $c^\infty$ topology (i.e. the final topology induced by all smooth curves). If these Fréchet spaces are finitedimensional and the sequence is strictly increasing, then Proposition 4.26 (iii), pp. 45 entails that the $c^\infty$ (and hence the Kelley) topology is not a vector space topology. 
Oct 20 
comment 
Is every Montel locally convex vector space compactly generated?
Strongly inaccessible cardinals are somewhat beyond my current mathematical knowledge, but of course your explanation makes clear why the continuum hypothesis is not needed regardless of that. Glad I learned something new... Thanks! 
Oct 17 
comment 
Is every Montel locally convex vector space compactly generated?
I'm actually thinking of the MackeyUlam theorem; moreover, Ulam originally showed that $\mathbb R$ does not admit a Ulam measure using the continuum hypothesis. Are there stronger recent results in this respect? 
Oct 16 
accepted  Is every Montel locally convex vector space compactly generated? 
Oct 16 
comment 
Is every Montel locally convex vector space compactly generated?
That is an interesting counterexample. In this case, $X$ has the Mackey topology and its strong dual $\mathbb{R}^{\mathbb{R}}$ coincides as a vector space with its algebraic dual, hence $X$ is even bornological. Moreover, if one assumes that the continuum hypothesis is true, even $\mathbb{R}^{\mathbb{R}}$ itself is (ultra)bornological. 
Oct 15 
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Is every Montel locally convex vector space compactly generated?
@JochenWengenroth Are there counterexamples to the BanachDieudonné theorem if $X$ is no longer the dual of a Fréchet space? Can one not circumvent those if one assumes, say, that $X$ is (semi)Montel and bornological as in my above comment? 
Oct 15 
comment 
Is every Montel locally convex vector space compactly generated?
@JohannesHahn of course, you are right, I forgot about this difference. 
Oct 15 
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Is every Montel locally convex vector space compactly generated?
In view of that, if $X$ is bornological (hence quasibarrelled) and semiMontel (hence Montel), I'm willing to bet that $X$ is compactly generated. That would actually be enough for the purposes I have in mind... 
Oct 15 
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Is every Montel locally convex vector space compactly generated?
@JohannesHahn Well, it seems to me that these inclusions must be extended to the inclusions of the vector subspaces generated by each compact subset, otherwise we cannot guarantee that the final topology will be linear. Since we also want a locally convex topology, it suffices to consider absolutely convex compact subsets (for $X$ semiMontel, these are the bipolars of bounded subsets of $X$). The picture that seems to emerge is that, for $X$ semiMontel, the LCTVS topology generated by the inclusions of compact subsets is the bornologification of $X$. 
Oct 15 
revised 
Is every Montel locally convex vector space compactly generated?
corrected a small typo 
Oct 14 
asked  Is every Montel locally convex vector space compactly generated? 
Sep 19 
awarded  Nice Answer 
Sep 15 
comment 
Poincare lemma for nonsmooth differentiable forms
Hmm, that's really interesting... I'll have a look at Preiss's paper to see what goes wrong. 
Sep 15 
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Poincare lemma for nonsmooth differentiable forms
Thanks both to Jochen and Igor! The conclusion of Preiss's paper actually looks similar to that counterexample for the Laplacian in the book of GilbargTrudinger. That's very interesting. 
Sep 14 
revised 
Poincare lemma for nonsmooth differentiable forms
corrected statement 
Sep 14 
revised 
Poincare lemma for nonsmooth differentiable forms
Improved notation 
Sep 14 
revised 
Poincare lemma for nonsmooth differentiable forms
Inequality corrected and related amendments in the subsequent text 
Sep 14 
revised 
Poincare lemma for nonsmooth differentiable forms
Corrected typos in the first sequence, added reference to Igor Khavkine's answer 