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Pseudocompact spaces $$X$$ and $$df$$-spaces $$C_{c}(X)$$. (English) Zbl 1048.46029
Let $$X$$ be a completely regular Hausdorff space, and let $$C_c(X)$$ denote the space of all continuous real-valued functions on $$X$$, endowed with the compact-open topology. A locally convex space $$E$$ is called a df-space if it has a fundamental sequence of bounded sets and is $$c_0$$-quasibarrelled; i.e., every null sequence in the strong dual $$E'_b$$ is equicontinuous. The class of df-spaces properly contains Grothendieck’s class of (DF)-spaces. A locally convex space is locally complete if every bounded set is contained in a Banach disc.
In their main theorem, the authors prove that $$C_c(X)$$ is a df-space if and only if the strong dual $$C_c(X)'_b$$ is a Fréchet space, and that this holds if and only if $$X$$ is pseudocompact and the weak dual $$C_c(X)' [\sigma(C_c(X)', C_c(X))]$$ is locally complete or, equivalently, if and only if each regular Borel measure on $$X$$ has compact support. This is related to a result of J. M. Mazon [Bull. Soc. R. Sci. Liège 53, 225–232 (1984; Zbl 0561.46002)]. The (DF)-property for $$C_c(X)$$ had been characterized by S. Warner [Duke Math. J. 25, 265–282 (1958; Zbl 0081.32802)]. The question when $$C_c(X)$$ is a df-space was then raised by H. Jarchow [“Locally convex spaces” (Mathematische Leitfäden, B. G. Teubner, Stuttgart) (1981; Zbl 0466.46001)] who had introduced the term df-space.
The present authors provide an example of a $$C_c(X)$$ space which is a df-space, but not a (DF)-space. Any such $$C_c(X)$$ must be $$\ell^\infty$$-barrelled and not $$\aleph_0$$-barrelled, thus also answering an implicit question of H. Buchwalter and J. Schmets [J. Math. Pures Appl. 52, 337–352 (1973; Zbl 0268.46025)]. The article contains some other relevant results and examples.

##### MSC:
 46E10 Topological linear spaces of continuous, differentiable or analytic functions 46A04 Locally convex Fréchet spaces and (DF)-spaces 46A08 Barrelled spaces, bornological spaces 46A30 Open mapping and closed graph theorems; completeness (including $$B$$-, $$B_r$$-completeness) 54C35 Function spaces in general topology 54D30 Compactness
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