infinite dimensional reflexive subspace of $C[0,1]$

Question

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.

score 2 · Answer 1 · answered Dec 10 '18 at 08:53

2

$C[0,1]$ 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].

answered Dec 10 '18 at 08:53

Kavi Rama Murthy

311,013

Thanks for the answer and for the book tip. – elliptic Dec 11 '18 at 08:33

infinite dimensional reflexive subspace of $C[0,1]$

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