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If $A,B\in \mathscr{L_H}$ in the lattice of subspaces of a Hilbert space $\mathscr{H}$,
then is it always true that $$A\sqsubseteq B\ \&\ B\sqsubseteq A \implies A=B\ ~ ?$$

Or is there maybe an additional requirement(s) that $\mathscr{L_H}$ be the lattice of closed subspaces, or that $\mathscr{H}$ be separable, or etc? That $\mathscr{L_H}$ is atomic and atomistic with the covering property seems to suggest it's always true. But there also seem to be some non-Hausdorff topologies over $\mathscr{L_H}$, and convergent sequences in $T_0$ topologies don't always have unique limits, which I'm finding confusing with respect to this equality question. So I just want to confirm (or deny) that if I prove $A\sqsubseteq B\ \&\ B\sqsubseteq A$ in $\mathscr{L_H}$, then I can infer $A=B$.

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    $\begingroup$ Doesn't this just follow from the asymmetry property of partially ordered sets?...But of course, for this you first have to prove that the structure you're dealing with is indeed a lattice... Perhaps you can provide a reference you are following or so. $\endgroup$
    – Tobias Fünke
    Commented Jun 16 at 6:56
  • $\begingroup$ @TobiasFünke Thanks, yes, but some posets are also interpolative, i.e., $A\sqsubseteq B$ implies there exists $C$ such that $A\sqsubseteq C \sqsubseteq B$. And then, vice versa, we'd have $B\sqsubseteq D \sqsubseteq A$. So I'm finding all that kind of confusing as well. $\endgroup$
    – eigengrau
    Commented Jun 16 at 6:58
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    $\begingroup$ Well, again: You should provide a definition, at best with a source so we can understand what you are working on. $\endgroup$
    – Tobias Fünke
    Commented Jun 16 at 6:59
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    $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$
    – Qmechanic ♦
    Commented Jun 16 at 7:07
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    $\begingroup$ As far as I remember, in Beltrametti-Cassinelli’s book (the second author was a professor of mine), it is used the lattice of closed subspaces of a Hilbert space where the partial order relation is nothing but the inclusion (of sets) relation. Therefore your implication is obvious. I was not able to understand the definition of partial order relation from your first reference (chapter 4). The relevant topology on that lattice is the strong operator one (referring to orthogonal projectors associated to closed subspaces) ant it is Hausdorff. $\endgroup$
    – Valter Moretti
    Commented Jun 16 at 8:29

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