Bernstein space $B_\sigma$ as a Banach space

Bernstein space $B_\sigma$ consists of all exponential type, less than or equal to $\sigma$, entire functions bounded on $\mathbf R$. $B_\sigma$ equipped with a sup-norm is proved to be a non-separable Banach space non-isomorphic to $\ell_{\infty}$ but involving an isometric copy of $\ell_{\infty}$. $B_\sigma$ is proved to be non-complemented in $B_\rho$, $\sigma<\rho$; $B_\sigma$ is also proved to be isometric to a second dual of its subspace $B_\sigma^0$ consisting of functions tending to zero along $\mathbf R$. The coincidence of weak and norm convergence of sequences (Schur property) in the dual of $B_\sigma^0$ is proved.