(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The theorem about the closest point property says:

If A is a convex, closed subspace of a hilbert space H, then

[tex] \forall x \in H\,\, \exists y \in A:\,\,\,\, \| x-y\| = \inf_{a\in A}\|x-a\|[/tex]

I have to show that it is enough to show this theorem for x = 0 only, by using the isometry [itex]T_{x_0}(x) = x_0 + x [/itex].

3. The attempt at a solution

So I would have to show that [itex] \exists y \in A [/itex] such that [itex]\|y\| = \inf_{a\in A}\|a\|[/itex], that is if A contains an element with least length, than for any point x in H there is a point in A that is closest to x, than any other in A.

Then what?

Any hint is appreciated.

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# Homework Help: Hilbert Space: Closest point property

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