Inner Products and Norms
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 i
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Inner Products and Norms 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. Mathematical analysis relies on the exploitation of inequalities. The most fundamental is the Cauchy–Schwarz inequality, which is valid in every inner product space. The more familiar triangle inequality for the associated norm is then derived as a simple consequence. Not every norm comes from an inner product, and, in such cases, the triangle inequality becomes part of the general definition. Both inequalities retain their validity in both finitedimensional and infinite-dimensional vector spaces. Indeed, their abstract formulation exposes the key ideas behind the proof, avoiding all distracting particularities appearing in the explicit formulas. The characterization of general inner products on Euclidean space will lead us to the noteworthy class of positive definite matrices. Positive definite matrices appear in a wide variety of applications, including minimization, least squares, data analysis and statistics, as well as, for example, mechanical systems, electrical circuits, and the differential equations describing both static and dynamical processes. The test for positive definiteness relies on Gaussian Elimination, and we can reinterpret the resulting matrix factorization as the algebraic process of completing the square for the associated quadratic form. In applications, positive definite matrices most often arise as Gram matrices, whose entries are formed by taking inner products between selected elements of an inner product space. So far, we have focussed our attention on real vector spaces. Complex numbers, vectors, and functions also arise in numerous applications, and so, in the final section, we take the opportunity to formally introduce complex vector spaces. Most of the theory proceeds in direct analogy with the real version, but the notions of inner product and norm on complex vector spaces require some thought. Applications of complex vector spaces and their inner products are of particular significance in Fourier analysis, signal processing, and partial differential equations, [61], and they play an absolutely essential role in modern quantum mechanics, [54].
3.1 Inner Products The most basic example of an inner product is the familiar dot product n vi wi , v · w = v1 w1 + v2 w2 + · · · + vn wn =
(3.1)
i=1
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