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

Let [math]\displaystyle{ H=\{(x_1,x_2,...)| \sum x_n^2\lt \infty\} }[/math] and define [math]\displaystyle{ S^{\infty}=\{x\in H| ||x||=1\} }[/math]

Claim

[math]\displaystyle{ S^{\infty} }[/math] is contractible

Proof

Suppose [math]\displaystyle{ x=(x_1,x_2,...)\in S^{\infty} }[/math] then [math]\displaystyle{ \sum x_n^2=1 }[/math]

Define [math]\displaystyle{ F:S^{\infty}\times I\rightarrow S^{\infty} }[/math] by [math]\displaystyle{ F(x,t)=((1-t)x_1+t\sqrt{x_1^2+x_2^2},(1-t)x_2,x_3,x_4,...)/||((1-t)x_1+t\sqrt{x_1^2+x_2^2},(1-t)x_2,x_3,x_4,...)|| }[/math]