Abstract
The weak topology of a locally convex space (lcs) E is denoted by w. In this paper we undertake a systematic study of those lcs E such that (E, w) is (linearly) Eberlein–Grothendieck (see Definitions 1.2 and 3.1). The following results obtained in our paper play a key role: for every barrelled lcs E, the space (E, w) is Eberlein–Grothendieck (linearly Eberlein–Grothendieck) if and only if E is metrizable (E is normable, respectively). The main applications concern to the space of continuous real-valued functions on a Tychonoff space X endowed with the compact-open topology \(C_k(X)\). We prove that \((C_k(X), w)\) is Eberlein–Grothendieck (linearly Eberlein-Grothen—dieck) if and only if X is hemicompact (X is compact, respectively). Besides this, we show that the class of E for which (E, w) is linearly Eberlein–Grothendieck preserves linear continuous quotients. Various illustrating examples are provided.
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Acknowledgements
The authors would like to thank the referee for several very valuable comments that essentially strengthened the presented results. In the previous version of the paper Theorem 2.11 was proved under some additional assumption on X. With the help of Theorem 2.7 an idea of which was suggested by the referee, we obtained a simpler proof of that result, without any additional assumptions. Similarly, an idea of Theorem 3.8 is due to the referee. These facts led to significant improvements in the exposition.
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Jerzy Ka¸kol is supported by the GAČR project 20-22230 L and RVO: 67985840.
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Ka̧kol, J., Leiderman, A. When is a Locally Convex Space Eberlein–Grothendieck?. Results Math 77, 236 (2022). https://doi.org/10.1007/s00025-022-01770-w
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DOI: https://doi.org/10.1007/s00025-022-01770-w