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On weakly compact sets in $$C\left( X\right) $$ C X
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas ( IF 1.8 ) Pub Date : 2021-01-04 , DOI: 10.1007/s13398-020-00987-0
J. C. Ferrando , S. López-Alfonso

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.



中文翻译:

在$$ C \ left(X \ right)$$ CX中的弱紧集上

子集一个局部凸空间的ë被称为(相对)顺序地完成,如果每个柯西序列\(\左\ {X_ {N} \右\} _ {N = 1} ^ {\ infty} \)ë包含在收敛到一个点\(X \在A \) (点\(X \于E \) )。Asanov和Velichko证明,如果X是非常紧凑的,则\(C_ {p} \ left(X \ right)\)中的每个有界集都是相对紧凑的,而Baturov表明,如果X是Lindelöf \(\ Sigma \)空间,每个空间都在\(C_ {p} \ left(X \ right)\)中设置是整体式紧凑型。我们证明,如果X是Lindelöf \(\ Sigma \) -空间,则每个在((C){C_ {p} \ left(X \ right)\)\(C_ {w} \ left(X \ right)\),i。例如,在配备了弱拓扑的\(C_ {k} \ left(X \ right)\)中,(相对)Gul'ko紧凑。我们得到一些后果。

更新日期:2021-01-04
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