Is there a non-Riesz way to prove the existence of uniqueness of this norm-preserving linear extension on Hilbert space?

Question

$M$ is a linear subspace of Hilbert space $H$, $f$ is a bounded linear functional on $M$.

Proof that there exists a norm-preserving linear extension of $f$ to $H$ call $F$ such that $F(M^{\perp})=0$, and $F$ is unique.

I know the way to prove it using Riesz representation theorem like that:questions/4445560

And I am questing that is it possible to prove it without Riesz representation theorem, it is hard for me so I came here for help.

Thanks if you can give me any advice of any form.