- #1
P3X-018
- 144
- 0
Homework Statement
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].
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.