As in the title, imagine we are in a generic TVS $E$, adding if necessary other hypothes (as E locally convex to have a nontrivial topological dual). Le $E'$ be the topological dual of $E$. Let $M$ be a subspace of $E$ and define the orthogonal of $M$ as:
$$ M^{\perp}=\{f\in E'\ s.t.\ \langle f,x\rangle=0\ \forall x\in M\} $$
where by $\langle\cdot,\cdot\rangle$ we denote the duality pairing.
Suppose it holds:
$$P\!:\; x\in M^{\perp}\Rightarrow x=0\ in\ E'\subset M'$$
As it is well known, in an Hilbert space we have that if $M$ is an orthonormal subset then $P$ is necessary and sufficient for having $M$ span the space, in the sense of a Schauder basis. (In this case both the dual and the orthogonal are 'internalized' by Riesz representation theorem).
I wonder if we have some interesting, though possibly weaker, results about subsets respecting $P$ also in more general context, and if not why it is so. In a sense then, what is the gist of the property P.
Thanks in advance.
P.S. this other question of mine is somehow an example (though not 100% precise) Interpretation of lemma: $f\in L^{1}_{loc}(\Omega)\ st\ \int f\phi=0\: \forall\phi\in\mathcal{D}(\Omega)\ \Rightarrow f=0\ in\ L^{1}_{loc}(\Omega)$.