On precompact sets in spaces Cc(X)
-
Juan Carlos Ferrando
and
Jerzy Ka̧kol
Abstract.
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.
© 2013 by Walter de Gruyter Berlin Boston
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Articles in the same Issue
- Masthead
- Fixed points of asymptotically regular mappings in complex-valued metric spaces
- Coverings of internal groupoids and crossed modules in the category of groups with operations
- A new study on almost increasing and quasi-monotone sequences
- On precompact sets in spaces Cc(X)
- Weighted boundedness for Toeplitz type operators related to Riesz transforms of the Schrödinger operator
- Sums of absolutely nonmeasurable functions
- Symmetry of an integral equation with Bessel potential in â„ťn+
- Spanier spaces and the covering theory of non-homotopically path Hausdorff spaces
- On the maximal operator of Marcinkiewicz–Fejér means of the two-dimensional Walsh–Kaczmarz system
- Hofer type geometry on the space of Legendrian submanifolds of a contact manifold
- Initial problems for hyperbolic functional differential systems
- Approximation by complex q-Bernstein–Schurer operators in compact disks
- On some properties of conjugate trigonometric Fourier series
