0708-1300/the unit sphere in a Hilbert space is contractible: Difference between revisions
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'''Proof''' |
'''Proof''' |
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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)=(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)=(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)=(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 ... |
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applying the homotopy <math>F_1</math> in the time interval <math>[0,1/2]</math>, <math>F_2</math> in the interval <math>[1/2,3/4]</math>, <math>F_3</math> in <math>[3/4,5/6]</math> etc... |
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we get the desired contraction to the point <math>(1,0,0,...)</math>. |
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Revision as of 16:29, 2 November 2007
Let and define
Claim
is contractible
Proof
A way to see this is via the cellular structure of . If you can always contract along like moving contracting the equator along the surface of the earth.
Does this proof only works in separable Hilbert spaces? Is the unit ball in a non-separable Hilbert space contractible?