Another nice solution to a similar question is at http://katlas.math.toronto.edu/drorbn/index.php?title=0708-1300/the_unit_sphere_in_a_Hilbert_space_is_contractible:
Let H=L^2([0,1])$H=L^2([0,1])$ and define S^\infty = {x \in H : ||x||=1}$S^\infty = \{x \in H : \|x\|=1\}$.
Claim. S^\infty$S^\infty$ is contractible.
Proof. For any t \in [0,1]$t \in [0,1]$ and any f \in H$f \in H$ define f_t(x)= f$f_t(x)= f$ for 0<x<t$0<x<t$ and f_t(x)=1$f_t(x)=1$ for t<x<1$t<x<1$. Observe that t --> f_t/||f_t||$t \mapsto f_t/\|f_t\|$ is continuous and gives the desired retraction to the point f=1$f=1$.
