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][1]:

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 --> f\_t/||f_t|| is continuous and gives the desired retraction to the point f=1.


  [1]: http://katlas.math.toronto.edu/drorbn/index.php?title=0708-1300/the_unit_sphere_in_a_Hilbert_space_is_contractible