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- W4308234047 abstract "By using the Principle of Local Reflexivity (PLR), we prove that for every two Banach spaces $E$ and $X$ there exists a suitable ultrafilter $mathcal{U}$ such that $ mathcal{F}(E,X)^*,$ the dual space of the finite rank operators, can be isomorphically identified with certain quotient of the ultrapower space $(Ewidehat{otimes} X^*)_mathcal{U}$, of the projective tensor product space $Ewidehat{otimes} X^*.$ This generalizes the identity $mathcal B(E,X^{**})congmathcal B(E,X)^{**}$, where $E$ is finite-dimensional. We then serve our main result to improve some results on the reflexivity of $mathcal B(E,X)$, the space of all bounded linear operators, by showing that: if $mathcal B(E,X)$ is reflexive, then $mathcal B(E,X)=mathcal A(E,X)$, the space of all approximable operators. This particularly implies that, $mathcal B(E)$ is reflexive if and only if $E$ is finite-dimensional. Finally, as more by-products of the PLR, some generalizations of the classical Goldstine weak$^*$-density theorem are also included." @default.
- W4308234047 created "2022-11-09" @default.
- W4308234047 creator A5042344895 @default.
- W4308234047 creator A5087445391 @default.
- W4308234047 date "2021-11-30" @default.
- W4308234047 modified "2023-09-29" @default.
- W4308234047 title "The principle of local reflexivity and an extension of the identity $mathcal B(E,X^{**})congmathcal B(E,X)^{**}$" @default.
- W4308234047 doi "https://doi.org/10.48550/arxiv.2112.00099" @default.
- W4308234047 hasPublicationYear "2021" @default.
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