Let $X$ be a Banach space (not necessarily reflexive or Hilbert) and $X'$ its topological dual. The "evaluation" pairing gives rise to the relation $\perp \subset X' \times X$, so we get an induced antitone galois correspondence $S ↦ S^\perp: \text{Sub}(X) \to \text{Sub}(X')$ (the annihilator) vs. $T ↦ T_\perp: \text{Sub}(X') \to \text{Sub}(X)$ (the preannihilator), where $\text{Sub}(V)$ is the poset of vector subspaces of $V$.
From that, it immediately follows that both
- $S ↦ (S^\perp)_\perp$ and
- $T ↦ (T_\perp)^\perp$
are closure operators on $\text{Sub}(X)$ and $\text{Sub}(X')$, respectively. It is not hard to see that the first closure operator is precisely the topological closure (preannihilators are always closed in one direction and hahn Banach in the other).
But what about the second one, the annihilator of the preannihilator? Does it coincide with the strong or weak* closure, or is it something different entirely?
This question seems to imply that it is not the strong topological closure.
