Abstract
This survey paper collects some of older and quite new concepts and results from descriptive set topology applied to study certain infinite-dimensional topological vector spaces appearing in Functional Analysis, including Fréchet spaces, (LF)-spaces, and their duals, (DF)-spaces and spaces of continuous real-valued functions C(X) on a completely regular Hausdorff space X. Especially (LF)-spaces and their duals arise in many fields of Functional Analysis and its applications, for example in Distributions Theory, Differential Equations and Complex Analysis. The concept of a realcompact topological space, although originally introduced and studied in General Topology, has been also studied because of very concrete applications in Linear Functional Analysis.
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Dedicated to Professor Manuel Valdivia, excellent professor and mathematical researcher, on the occasion of his 80th birthday.
The research for the first named author was (partially) supported by Ministry of Science and Higher Education, Poland, Grant no. NN201 2740 33 and for the both authors by the project MTM2008-01502 of the Spanish Ministry of Science and Innovation.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Ka̧kol, J., López-Pellicer, M. On realcompact topological vector spaces. RACSAM 105, 39–70 (2011). https://doi.org/10.1007/s13398-011-0003-0
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DOI: https://doi.org/10.1007/s13398-011-0003-0
Keywords
- Angelicity
- Baire and (b-) Baire-like
- Bornological
- Borel set
- C*-embedded
- Class \({\mathfrak{G}}\)
- (DF) space
- Distinguished space
- Fréchet–Urysohn
- k-Space
- K-Analytic
- (Weakly) Lindelöf (Σ)
- Locally convex space
- (Σ-)Quasi-Suslin space
- (Strongly) realcompact space
- (Compact) resolution
- Talagrand compact
- (Countable) tightness
- Trans-separable
- Weakly compact
- (WCG) space
- Web-bounded (compact)