TITLE: What is the meaning of the Riesz's lemma?
QUESTION [15 upvotes]: I am reading about the Riesz's lemma but I am struggling to understand the real meaning of it. I have read different proofs of the lemma and even though I understood the proofs I am still not sure what the lemma means or what are its consequences or why its important. Is there a simple explanation of a graphical representation of the lemma?
Thanks.

REPLY [8 votes]: Consider first a Hilbert space with a closed subspace $Y$. Then there exists the orthogonal complement $Y^{\bot}$. In particular, one can find a unit vector $x$ such that $x\bot Y$, or equivalently, $|x-y|\ge 1$ for all $y\in Y$.
The situation for Banach spaces (and normed spaces in general) is more complicated. One cannot find a unit "normal" vector to a closed subspace $Y$, i.e. to get $|x-y|\ge 1$ for all $y\in Y$ is impossible. However, it is possible to find an "almost normal" unit vector (in fact, arbitrary close to "being normal") in the sense that $|x-y|>r$, $\forall y\in Y$, where $r<1$ can be chosen arbitrary close to one.
Riesz's lemma says that for any closed subspace $Y$ one can find "nearly perpendicular" vector to the subspace.