0708-1300/the unit sphere in a Hilbert space is contractible

Let $H=L^{2}[0,1]$ and define $S^{\infty }=\{x\in H|||x||=1\}$

Claim

$S^{\infty }$ is contractible

Proof

For any $t\in [0,1]$ and any $f\in H$ define $f_{t}(x)=f$ for $0\leq x\leq t$ and $f_{t}(x)=1$ for $t<x\leq 1$. Observe that $t\rightarrow f_{t}/||f_{t}||$ is continuous and gives the desired retraction to the point $f=1$.

This proof only works in separable Hilbert spaces? Is the unit ball in a non-separable Hilbert space contractible?