reflexivity of space of bounded linear operators

Question

Is it true that if $X,Y$ are reflexive normed spaces then the space of bounded linear operators $B(X,Y)$ is reflexive ? If this statement is an immediate consequence of some known theorem where can I find this theorem ?

Thanks for any help.

Answer 1

In general, $B(X,Y)$ is not reflexive. A simple typical example is $B(\ell^2,\ell^2)$ because one easily checks that $$\ell^\infty\to B(\ell^2,\ell^2),\, x\mapsto m_x$$ where $m_x(y)=xy=(x_ny_n)_{n\in\mathbb N}$ is an isometry (for the $\sup$-norm on $\ell^\infty$ and the operator norm on $B(\ell^2,\ell^2)$).