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

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Let H=\{(x_1,x_2,...)| \sum x_n^2<\infty\} and define S^{\infty}=\{x\in H| ||x||=1\}

Claim

S^{\infty} is contractible

Proof

A way to see this is via the cellular structure of S^{\infty}. If S^{\infty}=C_0 C_1 ... you can always contract C_k along C_{k+1} like moving contracting the equator along the surface of the earth.

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