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Let $X$ be a (Tychonoff) topological space. Consider $C\left(X\right)$ being a topological vector space of all continuous scalar-valued functions with the compact-open topology.

Assume that $Y$ is a barreled space and $T$ is a linear map from $Y$ into $C\left(X\right)$ having a closed graph.

Could you please advice me the conditions on $X$, which would allow to apply the Closed Graph Theorem?

Clearly the assumption that $C\left(X\right)$ is an F-space (which is equivalent for $X$ to be hemicompact and compactly generated) is sufficient, but these two conditions are too restrictive.

For instance, it would be wonderful, if the second assumption alone was enough.

Thank you.

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    $\begingroup$ have you tried bourbaki? $\endgroup$
    – Koushik
    Jun 19, 2014 at 6:16
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    $\begingroup$ This should be a comment but I don't have enough brownie points. Firstly, I don't understand the downvote(s)---seems an interesting and highly non-trivial question to me. I would suggest the monograph "Topics in locally convex spaces" by Valdivia which has several delicate versions of the closed graph theorems in chapter 4 but this would then leave you with the task of pinpointing topological properties of $X$ which dualise to the corresponding ones for the lcs $C(X)$.Other sources for less standard CGT's are the classical text of Gottfried Köthe and the works of Pták and de Wilde. $\endgroup$
    – blackburne
    Jun 19, 2014 at 14:03
  • $\begingroup$ In fact my question is exactly about "pinpointing topological properties of X which dualise to the corresponding ones for the lcs $C\left(X\right)$. For example in Kelly-Namioka, they give few sufficient conditions for Closed Graph Theorem to hold, including hypercompleteness and full completeness, but I have no idea which properties of $X$ are needed for this. $\endgroup$
    – erz
    Jun 19, 2014 at 22:41
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    $\begingroup$ There are Springer Lecture Notes of Jean Schmets about locally convex properties of $C(X)$ (SLN 519, Espaces de fonctions continue, 1976). $\endgroup$
    – Jochen Wengenroth
    Jun 20, 2014 at 13:14
  • $\begingroup$ Once again, this is a comment not an answer but the system doesn't let me even comment on my own comments. Want to repeat the fact that I am completely gobsmacked by the fact that a question of this subtlety not only received downvotes but there were even attempts to close it (notabene by someone whose field of expertise is group theory). $\endgroup$
    – blackburne
    Jun 20, 2014 at 16:20

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