# Difference between revisions of "0708-1300/the unit sphere in a Hilbert space is contractible"

Let $H=L^2[0,1]$ and define $S^{\infty}=\{x\in H| ||x||=1\}$
$S^{\infty}$ is contractible
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. Observe that $t\rightarrow f_t/||f_t||$ is continuous and gives the desired retraction to the point $f=1$.