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Locally solid topologies on spaces of vector-valued continuous functions. (English) Zbl 1068.46023
The authors investigate properties of the so-called locally solid topologies on spaces of continuous functions with values in a normed space, as well as relations between locally solid topologies, strict topologies and Dini topologies. Let \(X\) be a completely regular Hausdorff space. Let \(E\) be a real normed linear space and let \(C_b(X,E)\) denote the set of all bounded continuous functions defined on \(X\) and having values in \(E\). A subset \(H\) of \(C_b(X,E)\) is called solid if \(\| f_1(x)\| _E\leq \| f_2(x)\| _E\) for all \(x\in X\), \(f_1\in C_b(X,E)\), \(f_2\in H\) implies \(f_1\in H\). A linear topology on \(C_b(X,E)\) is locally solid when it has a local base of zero consisting of solid sets. First, a number of results are presented on basic properties of locally solid topologies. Next, the relationship between topological structures of \(C_b(X,E)\) and \(C_b(X,\mathbb R)\) is studied. Using these results, a relationship is established also between strict topologies on \(C_b(X,E)\) and \(C_b(X,\mathbb R)\). The results enable the authors to examine strict topologies on \(C_b(X,E)\) by means of those on \(C_b(X,\mathbb R)\). A typical result states that certain strict topologies on \(C_b(X,E)\) are locally solid. Last, \(\sigma \)-Dini and Dini topologies are involved. A locally convex solid topology is said to be a \(\sigma \)-Dini’s topology whenever \(\| f_n(x)\| \downarrow 0\) for all \(x\in X\) implies \(f_n\to 0\) for \(\tau \). The Dini topology is defined analogously, replacing a sequence by a net. One of the main results of the paper states that a certain strict topology is the finest \(\sigma \)-Dini topology on \(C_b(X,E),\) and the corresponding result for Dini topologies. Finally, the authors characterize \(\sigma \)-Dini and Dini topologies by means of their topological duals.
46E40 Spaces of vector- and operator-valued functions
46A03 General theory of locally convex spaces
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