In Hilbert space, Closed points in Convex subsets

Question

I recently started to study about Hilbert space and the following is a Lemma of approximation:

I understand the whole proof except for one part. Why is the infimum $d$ of $A$ defined? We know $A$ is a closed convex subset of $H$, does this imply it is also bounded?

I have the feeling , this is quite elementary but I've been stuck here. I would appreciate any help. Thanks in advance!!!

Answer 1

No, $A$ may not bounded. And it is not "the infimum of $A$", but the infimum of the set $\{\lVert x-a\rVert\,:\,a\in A\}$. The map $\lVert \bullet -x\rVert:A\to \Bbb R$ takes values in $[0,\infty)$. So $\inf_{a\in A}\lVert a-x\rVert$ exists and it is a non-negative real number.

Answer 2

No. If $A$ is a closed and convex subset of $H$, then $A$ need not to be bounded. Example: $A=H$.