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Weakly $${p}$$-Dunford Pettis sets in $$ {L_1(\mu ,X)}$$
Annals of Functional Analysis ( IF 1 ) Pub Date : 2020-10-23 , DOI: 10.1007/s43034-020-00091-9
Ioana Ghenciu

Sets in Banach spaces that are mapped into norm compact sets by operators $$T:X\rightarrow \ell _p$$ (called weakly p-Dunford Pettis sets), for $$1< p< \infty $$ , are studied in arbitrary Banach spaces X and in the space $$L_1(\mu , X)$$ of Bochner integrable functions. Sufficient conditions for a subset of $$L_1(\mu , X)$$ to be a weakly p-Dunford Pettis set are given. It is shown that if $$X^*\in C_{p}$$ , and K is a bounded and uniformly $$L_p$$ -integrable subset of $$L_p(\mu , X)$$ , then K is a weakly p-DP set in $$L_1(\mu , X)$$ .

中文翻译:

弱 $${p}$$-Dunford Pettis 集于 $$ {L_1(\mu ,X)}$$

Banach 空间中的集合被操作符 $$T:X\rightarrow \ell _p$$(称为弱 p-Dunford Pettis 集)映射到范数紧集,对于 $$1< p< \infty $$ ,被任意研究Banach 空间 X 和 Bochner 可积函数的空间 $$L_1(\mu , X)$$。给出了 $$L_1(\mu , X)$$ 的子集是弱 p-Dunford Pettis 集的充分条件。结果表明,如果 $$X^*\in C_{p}$$ ,并且 K 是 $$L_p(\mu , X)$$ 的有界且一致的 $$L_p$$ -可积子集,则 K 是在 $$L_1(\mu , X)$$ 中设置的弱 p-DP。
更新日期:2020-10-23
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