The principle of local reflexivity and an extension of the identity $\mathcal B(E,X^{**})\cong\mathcal B(E,X)^{**}$

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 $(E\widehat{\otimes} X^*)_\mathcal{U}$, of the projective tensor product space $E\widehat{\otimes} X^*.$ This generalizes the identity $\mathcal B(E,X^{**})\cong\mathcal B(E,X)^{**}$, 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.