The aim of this work is intended to present a unified, mostly self-contained, survey of the theory of integral representation of bounded operators in topological spaces. It is written for the graduate students who are familiar with abstract measure and integration, and also with the main topics of Banach spaces and topological vector spaces. A lot of work has been done on various extensions of this theorem and it is still the object of many investigations. The objective of this work is a presentation of some prominent generalizations of Riesz Theorem, frequently used in the literature. Each representation theorem considered here has its own integration process which goes through duality of function spaces. We will essentially be concerned by the following settings: In the first part, we consider Banach spaces X, Y, and form the Banach space C(S, X) of all continuous functions from the compact space S into X, equipped with the uniform norm. In the second part, the objective is to go beyond the Banach space setting, to X a topological vector space context.

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