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# Development of the Fundamental Notions of Functional Analysis

## Overview

In the early twentieth century, mathematicians learned to give a geometrical interpretation to sets of functions that met certain overall conditions. This interpretation allowed mathematicians to assign a norm or "length" to each function in the set and to provide a measure of how much two functions differed from each other. One set of conditions on the functions in a set defined a Hilbert space, which could be treated as a vector space of infinite dimensions. Somewhat more general conditions were allowed in a Banach space. Both types of function spaces are of great importance in modern applied mathematics, and the ideas of Hilbert space play a particularly important role in quantum physics.

## Background

At the close of the nineteenth century many mathematicians believed that the theory of sets, introduced by the Russian-born German mathematician Georg Cantor (1845-1918), would provide a reliable and unifying basis for all mathematics. While not all mathematicians shared this view, it was embraced and popularized by the great German mathematician, David Hilbert (1862-1943), who in 1926 wrote, "No one shall expel us from the paradise which Cantor has created for us." By that time a small army of mathematicians had been at work, establishing a rigorous basis for set theory itself and providing set theoretic interpretations for algebra, geometry, the calculus, and the theory of numbers.

The most powerful notion used in applying set theory to other areas of mathematics is that of a mapping between sets. If A and B are sets, then a mapping from A to B associates with each member of A, or a subset of A, a particular member of B. A function, like y (x) = 7x + 2 can be thought of as a mapping from the set of real numbers into itself, since for every real number assigned to x a real number y is produced. The function y (x) = sin(x) is then a mapping from the set of real numbers to the set of real numbers between -1 and 1.

Another important notion is that of ordered pair—that is, a pair of elements, with the first drawn from a set A and the second from a set B. A function can then be thought of as a set of ordered pairs. The set of ordered pairs of real numbers can be taken as a description of the plane assumed in Euclidean geometry. The set of pairs of ordered pairs of real numbers, that is, the set of objects of the form ((a,b), (c,d)), where a, b , c, and d are each real numbers, then describes the set of directed line segments that can be drawn in the plane. Also, the length of each directed line segment, which we could write as length ((a,b), (c,d)), as determined by the Pythagorean theorem, could be thought of as a mapping from this set to the set of nonnegative real numbers. Likewise, the cosine of angle between the two line segments, which we can write as cos ((a,b), (c,d)), could be thought of as a mapping between the set of pairs of directed line segments and the real numbers between 1 and -1.

A number of mathematicians, including the great German mathematician Bernhard Riemann (1826-1866), had, as early as 1851, speculated that a geometrical interpretation could be given to sets of functions. The first systematic effort in this direction was provided in 1906 by French mathematician Maurice Frechet (1878-1973) in his doctoral thesis at the University of Paris. Frechet had first to concern himself with the idea of the limit of a sequence of points, so that he could speak of sets of functions as being complete if they contained the limit of every sequence of functions. Such a complete set of functions, then, would be called a metric space if there existed a mapping, which could be called the called the metric, between the set of all ordered pairs of functions and the set of nonnegative real numbers that had three important characteristics of length, namely:

metric (f,g) = metric (g,f) and is always nonnegative.

metric (f,g) = 0 if and only if f = g.

metric (f,g) + metric (g,h) is always greater or equal to metric (f, h).

The third requirement is called the triangle inequality because the sum of the lengths of any two sides of a triangle is always greater than the length of the remaining side.

David Hilbert became interested in sequences of functions as he began to consider the problem of expressing a given function as the sum of an infinite set of other functions. Hilbert's interest was ultimately a further development of the discovery by French engineer Jean-Baptiste Joseph Fourier (1968-1830) that any periodic function could be expressed as the sum of sine and cosine functions of the same period and whole number multiples of that period. Mathematicians had generalized Fourier's ideas to encompass other sets of functions, now called orthogonal functions, that could be used to express an arbitrary function defined over an interval of real numbers. Thus, given a set of orthogonal functions, any other function could be described over the same region by an infinite series of numbers, each of which described the contribution of the corresponding orthogonal function.

The fact that functions could be described by infinite series of numbers suggested that the set of all functions over a given interval could be treated as a vector space with an infinite number of dimensions. The implications of this important discovery were worked out primarily by German mathematician Erhardt Schmidt (1876-1959). Schmidt introduced the notion of an inner product space of functions. Such a space is characterized by a mapping, called the inner product, from the set of all pairs of functions to the set of real numbers. The inner product of a function with itself was always positive and could be taken as the square of a metric for the space. The inner product of two different functions could be given a geometrical interpretation as well, revealing how similar or "parallel" the two functions are. Despite the important role played by Schmidt, inner product spaces are usually called Hilbert spaces today.

Polish mathematician Stefan Banach (1892-1945) is generally credited with taking the most general approach to function spaces. Banach introduced the notion of a normed linear space, where the term norm denotes a generalized notion of length, and the space need not have an inner product. Such spaces, which include all Hilbert spaces as well as many others, are now called Banach spaces.

## Impact

The field of mathematics introduced by Frechet, Hilbert, and Banach is termed functional analysis today. The name is slightly inaccurate, as it reflects an earlier focus on functionals, which are mappings between sets of functions and the set of real numbers. The evaluation of a function for one value is a functional, as are the operations in calculus of taking the definite integral of a function over an interval and evaluating a derivative at a point. The theory of function spaces includes a number of results important in dealing with functionals, but it is now valued more as a general theory of the transformations that are possible among functions belonging to a Banach space.

Functional analysis is an area of mathematics that has been strongly influenced by the physical sciences, going back to the work of Fourier, whose studies on the expression of functions as sums of other functions appear in a work on the flow of heat, a subject of great importance to early nineteenth-century engineers. Another principle that had drawn the attention of mathematicians was the principle of least action in mechanics, stating that between any two points in space and time, a particle would follow that path that minimized a linear functional, the action integral, of the path.

The greatest interaction of functional analysis with physical sciences, however, occurred in the development of the quantum theory of matter in the 1920s. Beginning in 1900 physicists had discovered that many aspects of the behavior of matter could only be explained by assuming that certain mechanical properties could only take on certain values. Thus, the angular momentum of any subatomic particle as measured about a given axis could only be a whole number multiple of a fundamental value. Further, it was discovered that material particles, like the electron, had wavelike properties, and that their behavior in atoms could better be understood by treating the electrons as standing waves. Finally, it was discovered that the act of measurement of a given property would generally change the state of the system.

As was determined primarily by Hungarian-born mathematician John von Neumann (1903-1957), the allowable states of a quantum mechanical system can be viewed as represented by functions constituting a Hilbert space. The physically observable quantities such as energy and angular momentum are then each represented by transformations, called linear operators, that map the Hilbert space into itself. The effect of a measurement is to convert the function representing the system before the measurement into one of a selected number of states for which the representative function is simply multiplied by a real number under the transformation. Such functions are called eigenfunctions of the transformation. The number is the value of the physical quantity that is observed measured, and the probability of each possible value is given by the inner product of the original state function and the corresponding eigenfunction. Thus, Hilbert space theory provides the needed mathematical basis for the study of quantum phenomena.

DONALD R. FRANCESCHETTI