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Operator theory and functional analysis have a long tradition, initially being guided by problems from mathematical physics and applied mathematics. Much of the work in Banach spaces from the 1930's onwards resulted from investigating how much real (and complex) variable function theory might be extended to futions taking values in (function) spaces or operators acting in them. Many of the first ideas in geometry, basis theory and the isomorphic theory of Banach spaces have vector measure-theoretic origins and can be credited (amongst others) to N. Dunford, I.M. Gelfand, B.J. Pettis and R.S. Phillips. Somewhat later came the penetrating contributions of A.Grothendieck, which have pervaded and influenced the shape of functional analysis and the theory of vector measures/integration ever since. Today, each of the areas of functional analysis/operator theory, Banach spaces, and vector measures/integration is a strong discipline in its own right. However, it is not always made clear that these areas grew up together as cousins and that they had, and still have, enormous influences on one another. One of the aims of this monograph is to reinforce and make transparent precisely this important point.

Set functions. --- Linear operators. --- Function spaces. --- Functional analysis. --- Integral operators. --- Spaces, Function --- Functional analysis --- Linear maps --- Maps, Linear --- Operators, Linear --- Operator theory --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Functions, Set --- Functions of real variables --- Set theory --- Operators, Integral --- Integrals --- Global analysis (Mathematics). --- Operator theory. --- Analysis. --- Operator Theory. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical analysis. --- Analysis (Mathematics). --- 517.1 Mathematical analysis --- Mathematical analysis