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On weakly compact sets in \(C\left( X\right) \)


A subset A of a locally convex space E is called (relatively) sequentially complete if every Cauchy sequence \(\left\{ x_{n}\right\} _{n=1}^{\infty }\) in E contained in A converges to a point \(x\in A\) (a point \(x\in E\)). Asanov and Velichko proved that if X is countably compact, every functionally bounded set in \(C_{p}\left( X\right) \) is relatively compact, and Baturov showed that if X is a Lindelöf \(\Sigma \)-space, each countably compact (so functionally bounded) set in \( C_{p}\left( X\right) \) is a monolithic compact. We show that if X is a Lindelöf \(\Sigma \)-space, every functionally bounded (relatively) sequentially complete set in \(C_{p}\left( X\right) \) or in \(C_{w}\left( X\right) \), i. e., in \(C_{k}\left( X\right) \) equipped with the weak topology, is (relatively) Gul’ko compact. We get some consequences.

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This work was supported for the first named author by the Grant PGC2018-094431-B-I00 of Ministry of Science, Innovation and Universities of Spain.

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Correspondence to S. López-Alfonso.

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Ferrando, J.C., López-Alfonso, S. On weakly compact sets in \(C\left( X\right) \). RACSAM 115, 38 (2021).

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  • Lindelöf \(\Sigma \)-space
  • Realcompact space
  • \(\mu \)-Space
  • Sequentially complete set

Mathematics Subject Classification

  • 54C35
  • 54C05
  • 46A50