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

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?