Abstract
This chapter discusses how the theory of vector measures can be applied to derive universal properties of mappings on certain function spaces and how the theory of vector measures is connected with the theory of compact and weakly compact mappings on these spaces. The chapter discusses linear mappings. The chapter presents the following three results of A. Grothendieck's theory of [weakly] compact operators on C(S): (1) every linear bounded mapping T from C(S) into an L-space is weakly compact, (2) C(S) has the strict Dunford–Pettis property, that is, every linear weakly compact mapping from C(S) into another Banach space sends weak Cauchy sequences into strong Cauchy sequences, and (3) if S is Stonian, then any linear bounded mapping from C(S) into a separable Banach space is weakly compact.
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