Chapter 1 Basic concepts in the geometry of Banach spaces

Abstract

This chapter explains the basic concepts in the geometry of Banach spaces. Banach spaces will have either real or complex scalars. When the scalar field matters, the scalar field is mentioned explicitly, but in the notation for special spaces the scalars are not specified. Operators between Banach spaces are bounded and linear. An invertible operator T is called an “isomorphism.” Two norms on a vector space are called “equivalent” if the identity operator on X (with the two given norms) is an isomorphism. A projection is an idempotent operator. A subspace Z of X is said to be complemented if there is a projection from X onto Z . Probability theory has had a profound impact on Banach space theory. Consequently, it is used in places probabilistic terminology. The modern interpolation theory and a few applications to Banach space theory are also discussed in this chapter.