We know that $C[0,1]$, the space of functions continuous on interval $[0,1]$ equipped with maximum norm is not reflexive. Is there any infinite dimensional reflexive subspace of $C[0,1]$ or every infinite dimensional subspace is neccessarily non-reflexive ?
Thank you for your suggestions.
$C[0,1]$ is a universal space for the collection of separable Banach spaces in the sense any separable Banach space is isometrically isomorphic to a subspace of $C[0,1]$. This result is called the Banach - Mazur theorem. [You can find a proof in 'Geometric Functional Analysis and its Applications' by Holmes]. Hence there are lots of reflexive infinite dimensional subspaces. [ Embed $\ell ^{2}$ in $C[0,1]$, for example].