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

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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\leq1. 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?

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