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Filter-dependent versions of the uniform boundedness principle. (English) Zbl 1465.46002

A filter \(\mathcal F\) on \(\mathbb{N}\) is called: \(1^{\circ}\) a weak \(P^+\)-filter if for every decreasing sequence \((A_k)_k\) of sets in \(\mathcal F\) and for every \(\mathcal F\)-stationary set \(I\subseteq \mathbb{N}\) there exists an \(\mathcal F\)-stationary set \(B\subseteq I\) such that \(B\subseteq^*A_k\) for every \(k\); \(2^{\circ}\) a \(P^+\)-filter if for every sequence \((A_k)_k\) of \(\mathcal F\)-stationary sets, there exists an \(\mathcal F\)-stationary set \(B\) such that \(B\subseteq^*A_k\) for every \(k\); \(3^{\circ}\) a rapid\({}^+\) filter if for every \(\mathcal F\)-stationary set \(I\subseteq \mathbb{N}\) and every strictly increasing function \(f: \mathbb{N}\to \mathbb{N}\) there exists an \(\mathcal F\)-stationary set \(B\subseteq I\) such that the function enumerating \(B\) dominates \(f\), i.e., \(f\leq\eta_B\).
Let \(X, Y\) be topological vector spaces. The authors call a sequence \((T_i)_i\) in \(\mathcal L(X,Y)\) \(\mathcal F\)-stationary-equicontinuous if for every \(\mathcal F\)-stationary set \(I\subseteq \mathbb{N}\) and every \(0\)-neighbourhood \(V\) in \(Y\), there exists a \(0\)-neighbourhood \(U\) in \(X\) such that \((\exists J\in\mathcal F^+,J\subseteq I)\,(\forall i\in J)\,T_i[U]\subseteq V\). Further, the stationary uniform \(\mathcal F\)-boundedness principle is said to hold for continuous linear maps from \(X\) to \(Y\) when every pointwise \(\mathcal F\)-bounded sequence in \(\mathcal L(X,Y)\) is \(\mathcal F\)-stationary-equicontinuous, and it holds for \(X\) if it holds for the pair \(X,Y\) for every locally convex space \(Y\).
Among other results, the following is proved. The following statements are equivalent for a filter \(\mathcal F\) on \( \mathbb{N}\): \(1^{\circ}\) There exists an infinite-dimensional Banach space \(X\) such that the (stationary) uniform \(\mathcal F\)-boundedness principle holds for \(X\); \(2^{\circ}\) The (stationary) uniform \(\mathcal F\)-boundedness principle holds for every Fréchet space \(X\); \(3^{\circ}\) \(\mathcal F\) is both a rapid\({}^+\) and (weak) \(P^+\)-filter.

MSC:

46A04 Locally convex Fréchet spaces and (DF)-spaces
46B99 Normed linear spaces and Banach spaces; Banach lattices
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
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