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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.
##### MSC:
 46E40 Spaces of vector- and operator-valued functions 46A03 General theory of locally convex spaces
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