Difference between revisions of "0708-1300/the unit sphere in a Hilbert space is contractible"
From Drorbn
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− | A different way to see this is via the cellular structure of <math>S^{\infty}</math>. If <math>S^{\infy}= | + | A different way to see this is via the cellular structure of <math>S^{\infty}</math>. If <math>S^{\infy}=C_0 C_1 ...</math> you can always contract <math>C_k</math> along <math>C_{k+1}</math> like moving contracting the equator along the surface of the earth. |
Revision as of 11:05, 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 .
A different way to see this is via the cellular structure of . If Failed to parse (unknown function\infy): S^{\infy}=C_0 C_1 ...
you can always contract along like moving contracting the equator along the surface of the earth.