3 - Introduction to Functional Analysis

Abstract

This chapter provides an overview of functional analysis. It highlights various structures that can be defined on vector spaces. The chapter discusses the concept of a topological vector space, which is a vector space endowed with a topology compatible with the algebraic operations, that is, the topology makes vector addition and scalar multiplication continuous. In a Banach space, there is a notion of length of a vector, and in a Hilbert space, length is, in turn, determined by a dot product of vectors. Hilbert spaces are a natural generalization of finite-dimensional Euclidean spaces in the sense that many of the familiar geometric results in Rn carry over. A topological vector space is a vector space L with a topology such that addition and scalar multiplication are continuous. Two elements x and y in an inner product space L are said to be orthogonal or perpendicular if <x, y> = 0. According to Bessel's inequality theorem, if M is a subspace of Rn and x is any vector in Rn, x can be resolved into a component in M and a component perpendicular to M.