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 $H=L^2([0,1])$ 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 $f_t(x)= f$ for $0<x<t$ and $f_t(x)=1$ for $t<x<1$. Observe that $t \mapsto f_t/\|f_t\|$ is continuous and gives the desired retraction to the point $f=1$.
