Distinguished $$ C_{p}(X) $$ C p (
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Distinguished Cp (X) spaces J. C. Ferrando1
· J. Kakol ¸ 2,3 · A. Leiderman4 · S. A. Saxon5
Received: 4 November 2020 / Accepted: 11 November 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract We continue our initial study of C p (X ) spaces that are distinguished, equiv., are large subspaces of R X , equiv., whose strong duals L β (X ) carry the strongest locally convex topology. Many are distinguished, many are not. All L β (X ) spaces are, as are all metrizable C p (X ) and Ck (X ) spaces. To prove a space C p (X ) is not distinguished, we typically compare the character of L β (X ) with |X |. A certain covering for X we call a scant cover is used to find distinguished C p (X ) spaces. Two of the main results are: (i) C p (X ) is distinguished if and only if its bidual E coincides with R X , and (ii) for a Corson compact space X , the space C p (X ) is distinguished if and only if X is scattered and Eberlein compact. Keywords Distinguished space · Bidual space · Eberlein compact space · Fréchet space · strongly splittable space · Fundamental family of bounded sets · Point-finite family · G δ -dense subspace Mathematics Subject Classification 54C35 · 46A03
The first named author is supported by Grant PGC2018-094431-B-I00 of the Ministry of Science, Innovation ˇ Project and Universities of Spain. The research for the second named author is supported by the GACR 20-22230L and RVO: 67985840.
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J. C. Ferrando [email protected] J. K¸akol [email protected] A. Leiderman [email protected] S. A. Saxon [email protected]
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Centro de Investigación Operativa, Universidad Miguel Hernández, 03202 Elche, Spain
2
Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznan, Poland
3
Institute of Mathematics Czech Academy of Sciences, Prague, Czechia
4
Department of Mathematics, Ben-Gurion University of Negev, Beer Sheva, Israel
5
Department of Mathematics, University of Florida, Gainesville, FL 32611, USA 0123456789().: V,-vol
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1 Introduction Recall that a locally convex space E (embedded in its bidual E by means of the evaluation map) is semi-reflexive if E coincides algebraically with E, reflexive if it is semi-reflexive and the original locally convex topology of E coincides with β(E , E ), and distinguished if its strong dual E β is barrelled. Clearly, each reflexive space is semi-reflexive, and each semireflexive space is distinguished [26, 23.3 (4)]. In fact, (alternate definition [26, 23.7]), E is distinguished if and only if E is a large subspace of (E , σ (E , E )). Recall that a subspace F of a locally convex space G is a large subspace of G if every bounded set in G is contained in the closure in G of a bounded set in F, [32, Definition 8.3.22]. If X is a Tychonoff space and C p (X ) denotes the linear space C(X ) of all real-valued continuous functions defined on X equipped with the pointwise topology, it can be easily seen that C p (X ) is semi-reflexive if and only if C p (X ) is reflexive, if and
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