# STRICT TOPOLOGIES AND OPERATORS ON SPACES OF VECTOR-VALUED CONTINUOUS FUNCTIONS

• Nowak, Marian
• Published : 2015.01.01
• 39 5

#### Abstract

Let X be a completely regular Hausdorff space, and E and F be Banach spaces. Let $C_{rc}(X,E)$ be the Banach space of all continuous functions $f:X{\rightarrow}E$ such that f(X) is a relatively compact set in E. We establish an integral representation theorem for bounded linear operators $T:C_{rc}(X,E){\rightarrow}F$. We characterize continuous operators from $C_{rc}(X,E)$, provided with the strict topologies ${\beta}_z(X,E)$ ($z={\sigma},{\tau}$) to F, in terms of their representing operator-valued measures.

#### Keywords

spaces of vector-valued continuous functions;strict topologies;vector measures;integration operators

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2. Strongly bounded operators on Crc(X,E) with the strict topology βσ vol.27, pp.4, 2016, https://doi.org/10.1016/j.indag.2016.06.008