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**Complete spaces of \(p\)-adic measures.**
*(English)*
Zbl 1135.46045

Let \(\mathbb K\) be a complete non-Archimedean (n.a.) field, \(X\) a zero-dimensional Hausdorff topological space, \(E\) an n.a.Hausdorff locally convex space and \(C(X,E)\) be the space of all continuous functions from \(X\) to \(E\). Denote by \(C_b(X,E)\) (resp., \(C_{rc}(X,E)\)) the subspace of \(C(X,E)\) formed by all \(f\in C(X,E)\) for which \(f(X)\) is bounded (resp., relatively compact) in \(E\). The dual of \(C_{rc}(X,E)\) with the topology of uniform convergence is the space \(M(X,E')\) of all finitely additive \(E'\)-valued measures defined on the algebra \(K(X)\) of clopen subsets of \(X\). It turns out that some subspaces of \(M(X,E')\) are the duals of \(C(X,E)\) or of \(C_b(X,E)\) under certain locally convex topologies.

Continuing the investigations along these lines, the authors find some new types of such spaces and prove the completeness of the corresponding duals. For instance, under the hypothesis that \(E\) is a polar Fréchet space, the space \(M_s(X,E')\) for which \(ms\) is separable for all \(s\in E\) is complete with respect to the topology of uniform convergence on all equicontinuous subsets \(B\) of \(C(X,E)\) for which \(B(X)\) is a compactoid (here, \(ms(V)=m(V)s\), \(V\in K(X)\), \(s\in E\)). Other cases are related to the Banaschewski compactification of \(X,\) or concern the so-called strongly separable elements in \(M(X,E')\).

Continuing the investigations along these lines, the authors find some new types of such spaces and prove the completeness of the corresponding duals. For instance, under the hypothesis that \(E\) is a polar Fréchet space, the space \(M_s(X,E')\) for which \(ms\) is separable for all \(s\in E\) is complete with respect to the topology of uniform convergence on all equicontinuous subsets \(B\) of \(C(X,E)\) for which \(B(X)\) is a compactoid (here, \(ms(V)=m(V)s\), \(V\in K(X)\), \(s\in E\)). Other cases are related to the Banaschewski compactification of \(X,\) or concern the so-called strongly separable elements in \(M(X,E')\).

Reviewer: Stefan Cobzaş (Cluj-Napoca)

### MSC:

46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |

46E40 | Spaces of vector- and operator-valued functions |

46E27 | Spaces of measures |