Let $\mathcal{C}([0,1])$ the Banach space of continuous functions from $[0,1]$ to $\mathbb{C}$. The norm on $\mathcal{C}([0,1])$ is $f \mapsto \| f\|_{\infty}= \sup_{x \in [0,1]} |f(x)|$.

Is it possible that there is an homeomorphism $\phi$ from $\mathcal{C}([0,1])$ onto $H$, a Hilbert space ?

I have shown it's not possible if $\phi$ is supposed to be linear.


There is Anderson-Kadec Theorem: Each separable Frechet (=locally convex complete linear metric) space is homeomorphic to a Hilbert space.

Also there is a paper by Taras Banakh and Igor Zarichnyy, Topological groups and convex sets homeomorphic to non-separable Hilbert spaces, Central European J. of Math. 6:1 (2008), 77–86.


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