I was reading the wikipedia article of the representation theorem of Riesz–Fréchet.
see: https://en.wikipedia.org/wiki/Riesz_representation_theorem
In the proof of the theorem, they say
"Using Zorn's lemma or the well-ordering theorem it can be shown that there exists some non-zero vector $v\in M^{\perp}$"
$M$ denotes just the kernel of the functional $f\in\mathcal{H}^{\prime}$, which is assumed to be non-zero.
I am wondering why we need the Lemma of Zorn for this statement. I have had a look at different books about functional analysis and litarally nobody mentions this. Also a google search gives no result. I mean since we assume that $f$ is non-zero, we know that the kernel $M$ is not the whole Hilbert space and hence, the orthogonal complement $M^{\perp}$ is not empty and contains a non-trivial element by the projection theorem, which says that $\mathcal{H}=M\oplus M^{\perp}$.