Approximation properties and vector integrals in banach spaces

Abstract

This thesis is devoted to study of characterizations of approximation properties and the weak Radon-Nikodym property in Banach spaces and various properties of the Lebesgue space $L^P(G)$ of real valued measurable functions which are integrable with respect to a vector measure $\textsl{G}$. First, we establish new necessary and sufficient conditions for Banach spaces to have the approximation property. We modify the proof of Grothendieck for dual Banach spaces to have the approximation property. And we also add some properties and relations of a weak version of the approximation property. Secondly, we introduce Grothendieck`s method of relating the metric approximation property to tensor products, we establish another sufficient condition for dual spaces to have the metric approximation property. We also characterize the weak Radon-Nikodym property which is a candidate for another sufficient condition for dual spaces to have the metric approximation property. Finally, we investigate various properties of the Lebesgue space $L^P(G)$. And we also study conditions on $\textsl{G}$ which enables $L^1(G)$ to enjoy the property of the classical Lebesgue space. In particular, we show, by an example, that $L^1(G)$ does not have the approximation property in general.