The basic notions of the theory of Hilbert space are current in many parts of pure and applied mathematics, and in physics, engineering and statistics. They are well worth a place in any honours mathematics course, and Chapters 1 to 8 of this book aim to present them in a way accessible to undergraduate students. A course in Hilbert space is likely to be the last analysis course for many students, and it should therefore be able to stand on its own: it should not depend for its motivation on further study of abstract analysis, but should as far as possible have a value which is apparent either on aesthetic grounds or for its scientific or practical applications. For this reason I have included more historical and background material than is customary, and have omitted some of the major theorems about Banach spaces which are traditionally taught in introductory courses on functional analysis, but which are really more appropriate to students who will be pursuing operator theory further (the closed graph, Hahn–Banach and uniform boundedness theorems). The second half of the book describes two substantial applications. One of these is standard: the Sturm–Liouville theory of eigenfunction expansions, and its role in the solution of the partial differential equations of mathematical physics by the method of separation of variables. The other (in Chapters 12 to 16) is less common, but is nevertheless ideal for a final year course.

There is no shortage of essential uses of Hilbert space theory in mathematics and in pure and applied science, but setting up the background for any of them is lengthy. We shall content ourselves with two applications which illustrate Hilbert spaces in action in mainstream mathematics and also in science and engineering. In the next three chapters we shall develop the mathematical theory which justifies the method of ‘separation of variables’. This is one of the commonest approaches to the solution of linear partial differential equations, and is in routine use by scientists and working engineers. The topic thus has practical importance, and it also has historical significance since it was from the study of the differential and integral equations associated with such problems that functional analysis emerged.

Sturm–Liouville systems are second-order linear differential equations with boundary conditions of a particular type, and they usually arise from separation of variables in partial differential equations which represent physical systems. As an illustration we analyse small planar oscillations of a hanging chain.

Small oscillations of a hanging chain

A uniform heavy flexible chain of length L is freely suspended at one end and hangs under gravity. The chain is displaced slightly, in a vertical plane, and released from rest. The problem is to describe the subsequent motion of the chain. We are really concerned here with the mathematical analysis which is required, but for completeness let us give a brief heuristic derivation of the governing equations.

After Newton's success in giving a mathematical account of planetary motion, scientists were emboldened to attempt the description of many physical phenomena using the differential calculus. The ‘infinitesimal’ viewpoint gave a fruitful way of formulating laws governing everyday processes or their idealizations. Wave motion, fluid flow and the conduction of heat were analysed. Their description is harder than that of celestial motions, if only because the world of our experience has such an abundance of detail in comparison with the emptiness of space. Still, there is a miraculous feature of the fabric of the universe to hearten the mathematician: a great diversity of physical processes is well described by second-order linear partial differential equations. Much of functional analysis stems from the study of such equations.

Imagine that you are a pioneer of mathematical physics faced with a new class of equations of the general type Lf = g. Here g is supposed to be a known function of space and time variables, L is a linear differential operator and f is a function which is unknown but required to satisfy some initial or boundary conditions. Your first concern would be to devise a way of finding f. Even if you work out a bag of tricks that does well in practice, think of the difficulty of proving that a particular technique will always succeed for the class of equations under study.

The inner product spaces ℂn and ℓ2 share a further convenient property: informally speaking, any sequence in either of these spaces which looks convergent is convergent. To see what this means, recall the inner product space ℓ0 of Example 2.7. We saw there a sequence (xk) converging in ℓ2 but not in ℓ0. If we tried to carry out all our analysis in ℓ0 this phenomenon would definitely complicate matters: it would be like trying to do real analysis in ℚ instead of ℝ. Let us formulate the requirement of the existence of limits.

Definition Let (M,d) be a metric space. A sequence (xk) in M is a Cauchy sequence if, for every ε > 0, there exists an integer k0 such that k,l≥k0 implies that d(xk,xl) <ε.

(M,d) is a complete metric space if every Cauchy sequence in M converges to a limit in M.

Thus ℝ is a complete metric space with respect to its natural metric. So also is ℝ, for if (zk) is a Cauchy sequence in ℂ, then (Re zk) and (Im zk) are Cauchy sequences in ℝ. They thus have limits x,y∈ℝ, and we have zk → x + iy in ℂ.

Theorem ℂn (for n∈ℕ) and ℓ2 are complete metric spaces.

We recall our convention that metric terminology refers to the metric determined by the norm according to Theorem 2.3 for an inner product space.

A problem which commonly arises in statistics, physics and engineering is to find the function in a given class which approximates some known function as well as possible. To turn this vague notion into a well-posed mathematical question we need a numerical criterion of how good an approximation is. A popular choice is to look for a best ‘least squares’ fit. That is, we ask that the norm of the error function (the difference between the given function and the chosen approximation) be as small as possible, where the norm used is a Hilbert space norm. The advantage of this type of criterion is the obliging nature of Hilbert space geometry: such problems are usually comparatively easy to solve. Let us check this in a particular case.

Problem Given a function ϕ ∈ L2, find a function g∈H2 such that ∥ϕ - g∥L2 is minimized.

If we were to write out the definition of ∥ϕ - g∥ the reason for the expression ‘least squares’ would be apparent. In this instance the class of functions from which the approximation is to be chosen is a closed subspace of L2, and in this case the best approximation problem was essentially solved in Chapter 4. Let us recall the relevant facts and introduce some notation. If M is a closed subspace of a Hilbert space H and x∈H then, by Theorem 4.24, there exist y∈M, z∈M⊥ such that x = y + z.

What prospects are in view, what further peaks accessible from the high ground of Hilbert space? We trust our teachers, that the agony of learning they inflict on us is to good purpose, but resolution flags, and may perhaps be quickened by the contemplation of our goal.

Euclid was able to permit himself a lordly response to a pupil who asked about the uses of geometry. According to tradition Euclid summoned a servant and instructed him to give a penny to the pupil so that he should have profit from his studies. If your sympathies are with the recipient of the coin try reading A Mathematician's Apology (Hardy, 1940) for an updated and cogently argued presentation of the Euclidean attitude. Hardy firmly divides mathematics into two kinds: the useful but boring and ‘real mathematics’ which is beautiful but of no use whatever. In his inaugural lecture as professor at Oxford Hardy came close to saying that the value of mathematics is to absorb harmlessly the intellectual energies of very clever people who might otherwise be making weapons of destruction. Hardy was the most influential British mathematician of the day and his views took a long time to live down. Since then higher education has grown hugely, more colleges and new disciplines are vying for their share of the public purse, and we are all more conscious of the scrutiny of the Treasury and more careful of the public's perception.

Functional analysis is a branch of mathematics which uses the intuitions and language of geometry in the study of functions. The classes of functions with the richest geometric structure are called Hilbert spaces, and the theory of these spaces is the core around which functional analysis has developed. One can begin the story of this development with Descartes' idea of algebraicizing geometry. The device of using co-ordinates to turn geometric questions into algebraic ones was so successful, for a wide but limited range of problems, that it dominated the thinking of mathematicians for well over a century. Only slowly, under the stimulus of mathematical physics, did the perception dawn that the correspondence between algebra and geometry could also be made to operate effectively in the reverse direction. It can be useful to represent a point in space by a triple of numbers, but it can also be advantageous, in dealing with triples of numbers, to think of them as the co-ordinates of points in space. This might be termed the geometrization of algebra: it enables new concepts and techniques to be derived from our intuition for the space we live in. It is regrettable that this intuition is limited to three spatial dimensions, but mathematicians have not allowed this circumstance to prevent them from using geometric terminology in handling n-tuples of numbers when n ≥ 3. In the context of ℝn one routinely speaks of points, spheres, hyperplanes and subspaces.

Let us pause a while from the technicalities of spaces and operators and reflect on the place of functional analysis in the wider picture. On the strength of remarks earlier in the book about the genesis of the subject we should certainly expect that functional analysis would provide a framework in which to formulate, discuss and (sometimes) solve problems in the description of phenomena on the basis of classical physics. The hanging chain example of Chapters 9–11 is intended to illustrate this aspect. But this is far from being the only role of the subject. It often happens that concepts introduced and developed for one purpose subsequently provide the key to understanding a quite different set of problems. Basic Hilbert space theory is at once close enough to our geometric intuition to be readily assimilated and advanced enough to provide powerful tools for deepening our knowledge over a wide area of mathematics. It is routinely used in differential geometry, complex analysis, number theory – indeed, almost every branch. As we learn more mathematics we come to appreciate better the inter-relationship between its parts, but in view of the brevity of life no individual can fully grasp all the ways in which a particular intellectual current flows through the body of mathematics and on into science. We can still gain some understanding of the process by studying particular instances, as we form conceptions of the natural and social orders from the relatively few places, things and people we know.