Dunford--Pettis type properties of locally convex spaces
| Autor: | Gabriyelyan, Saak |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | In 1953, Grothendieck introduced and studied the Dunford--Pettis property (the $DP$ property) and the strict Dunford--Pettis property (the strict $DP$ property). The $DP$ property of order $p\in[1,\infty]$ for Banach spaces was introduced by Castillo and Sanchez in 1993. Being motivated by these notions, for $p,q\in[1,\infty]$, we define the strict Dunford--Pettis property of order $p$ (the strict $DP_p$ property) and the sequential Dunford--Pettis property of order $(p,q)$ (the sequential $DP_{(p,q)}$ property). We show that a locally convex space (lcs) $E$ has the $DP$ property iff the space $E$ endowed with the Grothendieck topology $\tau_{\Sigma'}$ has the weak Glicksberg property, and $E$ has the strict $DP_p$ property iff the space $(E,\tau_{\Sigma'}) $ has the $p$-Schur property. We also characterize lcs with the sequential $DP_{(p,q)}$ property. Some permanent properties and relationships between Dunford--Pettis type properties are studied. Numerous (counter)examples are given. In particular, we give the first example of an lcs with the strict $DP$ property but without the $DP$ property and show that the completion of even normed spaces with the $DP$ property may not have the $DP$ property. Comment: arXiv admin note: text overlap with arXiv:2402.08860 |
| Databáze: | arXiv |
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