Inner Products and Norms

Abstract

The geometry of Euclidean space is founded on the familiar properties of length and angle. The abstract concept of a norm on a vector space formalizes the geometrical notion of the length of a vector. In Euclidean geometry, the angle between two vectors is specified by their dot product, which is itself formalized by the abstract concept of an inner product. Inner products and norms lie at the heart of linear (and nonlinear) analysis, in both finite-dimensional vector spaces and infinite-dimensional function spaces. A vector space equipped with an inner product and its associated norm is known as an inner product space. It is impossible to overemphasize their importance for theoretical developments, practical applications, and the design of numerical solution algorithms.