9
$\begingroup$

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.

$\endgroup$
7
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.