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 1 : S ∞ × I → S ∞ {\displaystyle F_{1}:S^{\infty }\times I\rightarrow S^{\infty }} by F 1 ( 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_{1}(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},...)||}
F 2 : S ∞ × I → S ∞ {\displaystyle F_{2}:S^{\infty }\times I\rightarrow S^{\infty }} by F 2 ( x , t ) = ( ( 1 − t ) x 1 + t x 1 2 + x 3 2 , 0 , ( 1 − t ) x 3 , x 4 , x 5 , . . . ) / | | ( ( 1 − t ) x 1 + t x 1 2 + x 3 2 , 0 , ( 1 − t ) x 3 , x 4 , x 5 , . . . ) | | {\displaystyle F_{2}(x,t)=((1-t)x_{1}+t{\sqrt {x_{1}^{2}+x_{3}^{2}}},0,(1-t)x_{3},x_{4},x_{5},...)/||((1-t)x_{1}+t{\sqrt {x_{1}^{2}+x_{3}^{2}}},0,(1-t)x_{3},x_{4},x_{5},...)||}
F 3 : S ∞ × I → S ∞ {\displaystyle F_{3}:S^{\infty }\times I\rightarrow S^{\infty }} by F 3 ( x , t ) = ( ( 1 − t ) x 1 + t x 1 2 + x 4 2 , 0 , 0 , ( 1 − t ) x 4 , x 5 , x 6 , . . . ) / | | ( ( 1 − t ) x 1 + t x 1 2 + x 4 2 , 0 , 0 , ( 1 − t ) x 4 , x 5 , x 6 , . . . ) | | {\displaystyle F_{3}(x,t)=((1-t)x_{1}+t{\sqrt {x_{1}^{2}+x_{4}^{2}}},0,0,(1-t)x_{4},x_{5},x_{6},...)/||((1-t)x_{1}+t{\sqrt {x_{1}^{2}+x_{4}^{2}}},0,0,(1-t)x_{4},x_{5},x_{6},...)||}
and so on ...
applying the homotopy F 1 {\displaystyle F_{1}} in the time interval [ 0 , 1 / 2 ] {\displaystyle [0,1/2]} , F 2 {\displaystyle F_{2}} in the interval [ 1 / 2 , 3 / 4 ] {\displaystyle [1/2,3/4]} , F 3 {\displaystyle F_{3}} in [ 3 / 4 , 5 / 6 ] {\displaystyle [3/4,5/6]} etc...
we get the desired contraction to the point ( 1 , 0 , 0 , . . . ) {\displaystyle (1,0,0,...)} .
and define 
is contractible
then 
by 
by 
by 
in the time interval ![{\displaystyle [0,1/2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf40737bcb67d2b44e2827874fcbbc9e9d133ee1)
, 
in the interval ![{\displaystyle [1/2,3/4]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1cdfbb0bf7b7883367f10f7d0cf0c6b897c41a)
, 
in ![{\displaystyle [3/4,5/6]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94cb6091fd78124b0afcf1073464aa6528f2f850)
etc...
.
