This thesis is devoted to an investigation of various approximation properties in Banach spaces, relations between these properties, and various topologies on the space of operators.
First, we introduce various weak approximation properties, which are in general weaker than known approximation properties, and establish necessary and sufficient conditions for Banach spaces to have those properties. And we study inheritances of those properties from the dual space of a Banach space to the Banach space and vice versa.
Secondly, we study if one approximation property implies another. By providing elaborate examples we prove that some properties is strictly stronger than others. We also observe that for dual spaces or reflexive spaces more properties turn out to be equivalent.
Finally, we consider various locally convex vector topologies on the space of bounded linear operators on Banach spaces and investigate their topological properties, including compactness. We then apply the results to our study of the approximation properties.