Let H = { ( x 1 , x 2 , . . . ) | ∑ x n 2 < ∞ } {\displaystyle H=\{(x_{1},x_{2},...)|\sum x_{n}^{2}<\infty \}} and define S ∞ = { x ∈ H | | | x | | = 1 } {\displaystyle S^{\infty }=\{x\in H|||x||=1\}}
Claim
S ∞ {\displaystyle S^{\infty }} is contractible
Proof
Suppose x = ( x 1 , x 2 , . . . ) ∈ S ∞ {\displaystyle x=(x_{1},x_{2},...)\in S^{\infty }} then ∑ x n 2 = 1 {\displaystyle \sum x_{n}^{2}=1}
Define F : S ∞ × I → S ∞ {\displaystyle F:S^{\infty }\times I\rightarrow S^{\infty }} by F ( x , t ) = ( ( 1 − t ) x 1 + t x 1 2 + x 2 2 , ( 1 − t ) x 2 , x 3 , x 4 , . . . ) / | | ( ( 1 − t ) x 1 + t x 1 2 + x 2 2 , ( 1 − t ) x 2 , x 3 , x 4 , . . . ) | | {\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},...)||}
and define 
is contractible
then 
by 
