I should give a talk on something I'm working on, and I'd like to have a list, as complete as possible, of applications, in and out of functional analysis, of the following classical result by F. Riesz:
Riesz's lemma. Let $\mathcal V = (\mathbb V, \|\cdot\|)$ be a normed space over the normed field, $\mathcal K = (\mathbb K, |\cdot|)$, of real/complex numbers, $W$ a closed proper subspace of $\mathcal V$, and $\delta$ a real number with $0 < \delta < 1$. There then exists $x \in \mathcal V$ with $\|x\| = 1$ such that $\|x - y\| \ge \delta$ for all $y \in W$.
Classical applications of which I'm already aware:
- That the unit ball of a real/complex normed space $\mathcal V$ is compact iff $\mathcal V$ is finite-dimensional.
- The non-existence of certain measures for infinite-dimensional normed spaces.
- The spectral theorem for compact operators on a (complex) Banach space.
By the way, where can I find some focused discussion on the 2nd point of the above list? It seems to me that I read an excellent paper on the topic, some time ago, but I can't remember either the author(s), the journal, or other useful details, and I've started thinking that I dreamed of it.
Thanks in advance.