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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.

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
Full Text: DOI
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