III.37 Hilbert Spaces

The theory Of VECTOR SPACES [I.3 §2.3] and LINEAR MAPS [I.3 §4.2] underpins a large part of mathematics. However, angles cannot be defined using vector space concepts alone, since linear maps do not in general preserve angles. An inner product space can be thought of as a vector space with just enough extra structure for the notion of angle to make sense.

The simplest example of an inner product on a vector space is the standard scalar product defined on ⁿ, the space of all real sequences of length n, as follows. If v = (v₁, . . . , v_n) and w = (W₁,. . . , w_n) are two such sequences, then their scalar product, denoted 〈v,w〉, is the sum v_lw_l + v₂w₂ + · · · + v_n w_n. (For example, the scalar product of (3, 2, -1) and (1, 4, 4) is 3 × 1 + 2×4+ (-1) × = 7.)

Among the properties that the scalar product has are the following two.

(i) It is linear in each variable separately. That is, 〈 u + µ v,w 〉 = 〈u,w〉 + µ〈 v,w 〉 for any three vectors u, v, and w and any two scalars and µ, and similarly 〈 u, v + µw 〉 = 〈 u, v 〉 + µ〈 u, w 〉.

(it) The scalar product 〈 v,v 〉 of any vector v with itself is always a nonnegative real number, and is zero only if v is zero.

In a general vector space, any function 〈 v, w 〉 of pairs of vectors v and w that has these two properties is called an inner product, and a vector space with an inner product is called an inner product space. If the vector space has complex scalars, then instead of (i) one must use the following modification.

(i′) For any three vectors u, v, and w and any two scalarsand µ, 〈 u + µv,w 〉 = 〈u,w〉 + µ〈 v,w 〉 and That is, the inner product is conjugate-linear in the second variable.

The reason this has anything to do with angles is that in ² and ³ the scalar product of two vectors v and w works out as the length of v times the length of w times the cosine of the angle between them. In particular, since a vector v makes an angle of zero with itself, 〈v,v, 〉 is the square of the length of v.

This gives us a natural way to define length and angle in an inner product space. The length, or norm, of a vector v, denoted || v ||, is Given two vectors v and w, the angle between them is defined by the fact that it lies between 0 and π (or 180°) and its cosine is 〈 v, w 〉 / || v || || w || . Once length has been defined, we can also talk about distance: the distance d(v, w) between v and w is the length of their difference, or || v - w ||. This definition of distance satisfies the axioms for a METRIC SPACE [III.56]. From the notion of angle, we can say what it is for v and w to be orthogonal to each other: this simply means that 〈 v, w 〉 = 0.

The usefulness of inner product spaces goes far beyond their ability to represent the geometry of two- and three-dimensional space. Where they really come into their own is if they are infinite dimensional. Then it becomes convenient if they satisfy the additional property of completeness, which is briefly discussed at the end of [III.62]. A complete inner product space is called a Hilbert space.

Two important examples of Hilbert spaces are the following.

(i) l₂ is the natural infinite-dimensional generalization of ⁿ with the standard scalar product. It is the set of all sequences (a₁,a₂, a₃, . . .) such that the infinite sum | a² + | a₂|² + | a₂|² + · · · converges. The inner product of (a₁, a₂, a₃, . . .) and (b_l, b₂, b₃, . . .) is a_lb_l + a₂b₂ + a₃b₃ + · · · (which can be shown to converge by the CAUCHY SCHWARZ INEQUALITY [V.19]).

(ii) L₂[0, 2π] is the set of all functions f defined on the interval [0, 2π] of all real numbers between 0 and 2 π , such that the integral | f(x)|² d x makes sense and is finite. The inner product of two functions f and g in L₂ [ 0, 2π] is defined to be f(x)g(x) dx. (For technical reasons, this definition is not quite accurate, as a nonzero function can have norm zero, but this problem can easily be dealt with.)

The second of these examples is central to Fourier analysis. A trigonometric function is a function of the form cos(mx) or sin(nx). The inner product of any two different trigonometric functions is zero, so they are all orthogonal. Even more importantly, the trigonometric functions serve as a coordinate system for the space L₂[0,2π], in that every function f in the space can be represented as an (infinite) linear combination of trigonometric functions. This allows Hilbert spaces to model sound waves: if the function f represents a sound wave, then the trigonometric functions are the pure tones that are its constituent parts.

These properties of trigonometric functions illustrate a very important general phenomenon in the theory of Hilbert spaces: that every Hilbert space has an orthonormal basis. This means a set of vectors e_i with the following three properties:

• || e_i || = 1 for every i;
• 〈 e_i, e_j 〉 = 0 whenever i ≠ j; and
• every vector v in the space can be expressed as a convergent sum of the form Σ_iie_i.

The trigonometric functions do not quite form an orthonormal basis of L₂[0, 2π] but suitable multiples of them do. There are many contexts besides Fourier analysis where one can obtain useful information about a vector by decomposing it in terms of a given orthonormal basis, and many general facts that can be deduced from the existence of such bases.

Hilbert spaces (with complex scalars) are also central to quantum mechanics. The vectors of a Hilbert space can be used to represent possible states of a quantum mechanical system, and observable features of that system correspond to certain linear maps.

For this and other reasons, the study of LINEAR OPERATORS [III.50] on Hilbert spaces is a major branch of mathematics: see OPERATOR ALGEBRAS [IV.15].