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

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

Claim

[math]\displaystyle{ S^{\infty} }[/math] is contractible

Proof

For any [math]\displaystyle{ t\in[0,1] }[/math] and any [math]\displaystyle{ f\in H }[/math] define [math]\displaystyle{ f_t(x)= f }[/math] for [math]\displaystyle{ 0\leq x \leq t }[/math] and [math]\displaystyle{ f_t(x)=1 }[/math] for [math]\displaystyle{ t\lt x\leq1 }[/math]. Observe that [math]\displaystyle{ t\rightarrow f_t/||f_t|| }[/math] is continuous and gives the desired retraction to the point [math]\displaystyle{ f=1 }[/math].

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

The answer seems to be YES see Spheres in infinite-dimensional normed spaces are Lipschitz contractible