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

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Let H = L2[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 ft(x) = f for 0\leq x \leq t and ft(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

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