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A different way to see this is via the cellular structure of <math>S^{\infty}</math>. If <math>S^{\infty}=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. |
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A different way to see this is via the cellular structure of <math>S^{\infty}</math>. If <math>S^{\infty}=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. |
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This proof only works in '''separable''' Hilbert spaces. Is the unit ball in a non-separable Hilbert space contractible? |
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
you can always contract
along
like moving contracting the equator along the surface of the earth.
This proof only works in separable Hilbert spaces. Is the unit ball in a non-separable Hilbert space contractible?