Geodesic
Strict topologies and Banach-Steinhaus type theorems
Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 4, pp. 563-568.

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Let $X$ be a completely regular Hausdorff space, $E$ a real Banach space, and let $C_b(X,E)$ be the space of all $E$-valued bounded continuous functions on $X$. We study linear operators from $C_b(X,E)$ endowed with the strict topologies $\beta_z$ $(z=\sigma,\tau,\infty,g)$ to a real Banach space $(Y,\|\cdot\|_Y)$. In particular, we derive Banach-Steinhaus type theorems for $(\beta_z,\|\cdot\|_Y)$ continuous linear operators from $C_b(X,E)$ to $Y$. Moreover, we study $\sigma$-additive and $\tau$-additive operators from $C_b(X,E)$ to $Y$.
Classification : 46E10, 47A70, 47B38
Keywords: vector-valued continuous functions; strict topologies; locally solid topologies; Dini-topologies; strong Mackey space; $\sigma $-additive operators; $\tau $-additive operators 
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     title = {Strict topologies and {Banach-Steinhaus} type theorems},
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Nowak, Marian. Strict topologies and Banach-Steinhaus type theorems. Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 4, pp. 563-568. http://geodesic.mathdoc.fr/item/CMUC_2009__50_4_a5/