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Georgian Mathematical Journal

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In This Section
Volume 20, Issue 2 (May 2013)


On precompact sets in spaces Cc(X)

Juan Carlos Ferrando
  • Centro de Investigación Operativa, Universidad Miguel Hernández, 03202 Elche, Spain
  • Email:
/ Jerzy Ka̧kol
  • Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznań, Poland
  • Email:
Published Online: 2013-05-15 | DOI: https://doi.org/10.1515/gmj-2013-0022


We show that the fact that X has a compact resolution swallowing the compact sets characterizes those Cc(X) spaces which have the so-called 𝔊-base. So, if X has a compact resolution which swallows all compact sets, then Cc(X) belongs to the class 𝔊 of Cascales and Orihuela (a large class of locally convex spaces which includes the (LM) and (DF)-spaces) for which all precompact sets are metrizable and, conversely, if Cc(X) belongs to the class 𝔊 and X satisfies an additional mild condition, then X has a compact resolution which swallows all compact sets. This fully applicable result extends the classification of locally convex properties (due to Nachbin, Shirota, Warner and others) of the space Cc(X) in terms of topological properties of X and leads to a nice theorem of Cascales and Orihuela stating that for X containing a dense subspace with a compact resolution, every compact set in Cc(X) is metrizable.

Keywords: Compact resolution; space Cc(X); 𝔊-base; class 𝔊; K-analytic space; W-space; quasibarrelled space

About the article

Received: 2012-06-22

Revised: 2013-02-12

Accepted: 2013-04-12

Published Online: 2013-05-15

Published in Print: 2013-05-01

Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2013-0022. Export Citation

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