# Is $\mathcal{C}([0,1])$ homeomorphic to a Hilbert space?

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.