Chapter 5 - Vector Spaces, Hilbert Spaces, and the L2 Space

Abstract

This chapter reviews the definitions and properties of finite dimensional vector spaces based on an algebraic approach. Isomorphism is a one-to-one and onto mapping between two mathematical entities, which preserves the structure of those entities (whether the structure is connectivity in a graph or the binary operation in a group). The structure in a vector space consists of scalar multiplication and vector addition, and that is why isomorphism is defined from this perspective. The Fundamental Theorem of Finite-Dimensional Vector Spaces postulates that an n-dimensional vector space over scalar field R is isomorphic to Rn. Similarly, an n-dimensional vector space over scalar field C is isomorphic to Cn. As opposed to writing any element of the vector space as a finite linear combination of basis vectors, it is desirable to write any element as a series of basis vectors. Hilbert spaces have much of the associated geometry of familiar vector spaces because they are endowed with an inner product. A vector space with a norm is a normed linear space. A normed linear space, which is complete with respect to the norm is a Banach space. That is, a Banach space is a normed linear space in which Cauchy sequences converge.