The uniform boundedness principle and its corollaries from a logical point of view are statements of when one can swap quantifiers in Banach spaces. Take for instance the principle of condensation of singularities; if $V$ is a Banach space and $\{\phi_{ij} \}_{i, j \in \mathbb N} \subseteq V^*$ is a family of continuous linear functionals, then \begin{equation} (\forall i \in \mathbb N)(\exists f_i \in V)(\sup_{j \in \mathbb N} |\phi_{ij} (f_i)| = \infty) \iff (\exists f \in V)(\forall i \in \mathbb N)(\sup_{j \in \mathbb N} |\phi_{ij} (f_i)| = \infty). \end{equation} The forward is the non-trivial direction, and one of the standard proofs of the like use Baire category, where after one plays around with quantifiers, unions and intersections, we can show that in fact the set of $f \in V$ satisfying the property on the right is in fact generic (in a category sense) in $V$.

I was wondering if there's some kind of deeper interplay of logic, topology and functional analysis going on here, perhaps a general principle for the types of statements about Banach spaces in which we can swap quantifiers $(\forall \exists \iff \exists \forall)$. If so, perhaps this gives a logical heuristic for why the uniform boundedness principle holds; equivalence of weak boundedness and strong boundedness by itself is a surprising result. My intuition says something of the sort should exists, but I haven't been able to dig anything up on the subject.