Abstract:We introduce and study two new classes of Banach spaces, the so-called sequentially Right Banach spaces of order $p$, and those defined by the dual property, the sequentially Right$^*$ Banach spaces of order $p$ for $1\leq p\leq\infty$. These classes of Banach spaces are characterized by the notions of $L_p$-limited sets in the corresponding dual space and $R^*_p$ subsets of the involved Banach space, respectively. In particular, we investigate whether the injective tensor product of a Banach space $X$ and a reflexive Banach space $Y$ has the sequentially Right property of order $p$ when $X$ enjoys this property.
Keywords: Right topology; sequentially Right Banach space; pseudo weakly compact operator; Pe\l czy\'nski's property (V) of order $p$; limited $p$-converging operator; $p$-Gelfand--Phillips property; reciprocal Dunford--Pettis property of order~$p$
DOI: DOI 10.14712/1213-7243.2020.011
AMS Subject Classification: 46B20 47L05 46B25