vol 4: Glossary
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Hilbert space
Introduction
Hilbert space, named for David Hilbert, is a mathematical construction derived from ordinary three dimensional Euclidean space. It is the mathematical stage upon which we play the model of the physical world known as quantum mechanics.
We imagine that mathematical spaces exist in an abstract way, but in order to find our way around them, we need to intorduce systems of coordinates. Cartesian coordinates enable us to give each point in Euclidean space a name which we represent by a set of numbers called a vector. In one dimensional space (that is on a line) each point is represented by just one number, its distance from an arbitrary point called the origin, whose coordinate is 0. A point one unit to the right of the origin is called 1, a point two units to the left, -2 and so on.
Points in two dimensional space ("the cartesian plane") are specified by two numbers, and so on. Points in a space with a countable number of dimensions are specified by ordered sets or vectors with a countable number of elements or components. As the number of dimensions of space increases, the amount of information encoded in each point of the space also increases. Given enough dimensions, we can encode any spacetime history as a single point.
Each such vector may be understood as representing a series or discrete function, that is a mapping from the natural numbers 0, 1, 2, ... to another set of numbers, which may be natural, rational, real or complex. The number ex, for instance, is represented by the series 1 + x + x2/2 + ... + xn/n! + ... . In other words the value on dimension 1 is one, on dimension 2 is x, and so on. Since each point in Hilbert space may be used to represent an infinite series, we may think of Hilbert space as a space of functions, or function space. Quantum mechanics works by transforming vectors in Hilbert space according to certain rules.
Hilbert space is a metric space, which means that we can define a distance between two functions. This distance underlies the numerical results of quantum calculations.
Axioms
A Hilbert space is a vector space H over the field of complex (or real) numbers, together with a complex valued (or real valued) function (x, y) (called the inner product) defined on H X H, with the following properties:
1. (x.
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