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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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[1] Henri Buchwalter and Jean Schmets, Sur quelques propriétés de l’espace \?_\? (\?), J. Math. Pures Appl. (9) 52 (1973), 337 – 352 (French). · Zbl 0268.46025
[2] D. H. Fremlin, D. J. H. Garling, and R. G. Haydon, Bounded measures on topological spaces, Proc. London Math. Soc. (3) 25 (1972), 115 – 136. · Zbl 0236.46025 · doi:10.1112/plms/s3-25.1.115 · doi.org
[3] Hans Jarchow, Locally convex spaces, B. G. Teubner, Stuttgart, 1981. Mathematische Leitfäden. [Mathematical Textbooks]. · Zbl 0466.46001
[4] Jerzy Kąkol and Stephen A. Saxon, Montel (DF)-spaces, sequential (LM)-spaces and the strongest locally convex topology, J. London Math. Soc. (2) 66 (2002), no. 2, 388 – 406. · Zbl 1028.46003 · doi:10.1112/S0024610702003459 · doi.org
[5] J. Kakol, S. A. Saxon, and A. R. Todd, Weak barrelledness for \(C\left( X\right) \) spaces, preprint.
[6] W. Lehner, Über die Bedeutung gewisser Varianten des Baire’schen Kategorienbegriffs für die Funktionenräume \( C_{c}\left( T\right) \), Dissertation, Ludwig-Maximilians-Universität, München (1979). · Zbl 0468.46012
[7] José M. Mazón, On the df-spaces of continuous functions, Bull. Soc. Roy. Sci. Liège 53 (1984), no. 5, 225 – 232. · Zbl 0561.46002
[8] R. A. McCoy and Aaron R. Todd, Support spaces and \?(\?)-points in the Stone-Čech compactification of the integers, Southeast Asian Bull. Math. 20 (1996), no. 1, 11 – 18. · Zbl 0884.46018
[9] Jan van Mill, Sixteen topological types in \?\?-\?, Topology Appl. 13 (1982), no. 1, 43 – 57. · Zbl 0489.54022 · doi:10.1016/0166-8641(82)90006-2 · doi.org
[10] Peter D. Morris and Daniel E. Wulbert, Functional representation of topological algebras, Pacific J. Math. 22 (1967), 323 – 337. · Zbl 0163.36605
[11] Pedro Pérez Carreras and José Bonet, Barrelled locally convex spaces, North-Holland Mathematics Studies, vol. 131, North-Holland Publishing Co., Amsterdam, 1987. Notas de Matemática [Mathematical Notes], 113. · Zbl 0614.46001
[12] Stephen A. Saxon, Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology, Math. Ann. 197 (1972), 87 – 106. · Zbl 0243.46011 · doi:10.1007/BF01419586 · doi.org
[13] Stephen A. Saxon and L. M. Sánchez Ruiz, Dual local completeness, Proc. Amer. Math. Soc. 125 (1997), no. 4, 1063 – 1070. · Zbl 0870.46001
[14] Stephen A. Saxon and Ian Tweddle, Mackey ℵ\(_{0}\)-barrelled spaces, Adv. Math. 145 (1999), no. 2, 230 – 238. · Zbl 0951.46001 · doi:10.1006/aima.1998.1815 · doi.org
[15] Seth Warner, The topology of compact convergence on continuous function spaces, Duke Math. J. 25 (1958), 265 – 282. · Zbl 0081.32802
[16] Albert Wilansky, Modern methods in topological vector spaces, McGraw-Hill International Book Co., New York, 1978. · Zbl 0395.46001
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