0708-1300/the unit sphere in a Hilbert space is contractible: Difference between revisions
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Suppose <math>x=(x_1,x_2,...)\in S^{\infty}</math> then <math>\sum x_n^2=1</math> |
Suppose <math>x=(x_1,x_2,...)\in S^{\infty}</math> then <math>\sum x_n^2=1</math> |
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Define <math>F_1:S^{\infty}\times I\rightarrow S^{\infty}</math> by <math>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,...)||</math> |
Define <math>F_1:S^{\infty}\times I\rightarrow S^{\infty}</math> by <math>F_1(x,t)=(sgn(x_1)((1-t)|x_1|+t\sqrt{x_1^2+x_2^2}),(1-t)x_2,x_3,x_4,...)/||(sgn(x_1)((1-t)|x_1|+t\sqrt{x_1^2+x_2^2}),(1-t)x_2,x_3,x_4,...)||</math> |
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<math>F_2:S^{\infty}\times I\rightarrow S^{\infty}</math> by <math>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,...)||</math> |
<math>F_2:S^{\infty}\times I\rightarrow S^{\infty}</math> by <math>F_2(x,t)=(sgn(x_1)((1-t)|x_1|+t\sqrt{x_1^2+x_3^2}),0,(1-t)x_3,x_4,x_5,...)/||(sgn(x_1)((1-t)|x_1|+t\sqrt{x_1^2+x_3^2}),0,(1-t)x_3,x_4,x_5,...)||</math> |
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<math>F_3:S^{\infty}\times I\rightarrow S^{\infty}</math> by <math>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,...)||</math> |
<math>F_3:S^{\infty}\times I\rightarrow S^{\infty}</math> by <math>F_3(x,t)=(sgn(x_1)((1-t)|x_1|+t\sqrt{x_1^2+x_4^2}),0,0,(1-t)x_4,x_5,x_6,...)/||(sgn(x_1)((1-t)|x_1|+t\sqrt{x_1^2+x_4^2}),0,0,(1-t)x_4,x_5,x_6,...)||</math> |
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and so on ... |
and so on ... |
Revision as of 10:50, 2 November 2007
Let and define
Claim
is contractible
Proof
Suppose then
Define by
by
by
and so on ...
applying the homotopy in the time interval , in the interval , in etc...
we get the desired contraction to the point .