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- W4289366719 abstract "Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. We show that the following assertions are equivalent: (i) $L(X)$ is $ell_infty$-barrelled, (ii) $L(X)$ is $ell_infty$-quasibarrelled, (iii) $L(X)$ is $c_0$-barrelled, (iv) $L(X)$ is $aleph_0$-quasibarrelled, and (v) $X$ is a $P$-space. If $X$ is a non-discrete metrizable space, then $L(X)$ is $c_0$-quasibarrelled but it is neither $c_0$-barrelled nor $ell_infty$-quasibarrelled. We prove that $L(X)$ is a $(DF)$-space iff $X$ is a countable discrete space. We show that there is a countable Tychonoff space $X$ such that $L(X)$ is a quasi-$(DF)$-space but is not a $c_0$-quasibarrelled space. For each non-metrizable compact space $K$, the space $L(K)$ is a $(df)$-space but is not a quasi-$(DF)$-space. If $X$ is a $mu$-space, then $L(X)$ has the Grothendieck property iff every compact subset of $X$ is finite. We show that $L(X)$ has the Dunford--Pettis property for every Tychonoff space $X$. If $X$ is a sequential $mu$-space (for example, metrizable), then $L(X)$ has the sequential Dunford--Pettis property iff $X$ is discrete." @default.
- W4289366719 created "2022-08-02" @default.
- W4289366719 creator A5023391268 @default.
- W4289366719 date "2018-10-24" @default.
- W4289366719 modified "2023-09-23" @default.
- W4289366719 title "Locally convex properties of free locally convex spaces" @default.
- W4289366719 doi "https://doi.org/10.48550/arxiv.1810.10250" @default.
- W4289366719 hasPublicationYear "2018" @default.
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