| The theory of operators, created by V. Volterra, has as its object the study of functions defined on infinite-dimensional spaces. This theory has penetrated several highly important areas of mathematics in an essential way: suffice it to recall that the theory of integral equations and the calculus of variations are included as special cases within the main areas of the general theory of operators. In this theory the methods of classical mathematics are seen to combine with modern methods in a remarkably effective and quite harmonious way. The theory often makes possible altogether unforeseen interpretations of the theorems of set theory or topology. Thus, for example, the topological theorem on fixed points may be translated, thanks to the theory of operators (as has been shown by Birkhoff and Kellogg) into the classical theorem on the existence of solutions of differential equations. There are important parts of mathematics which cannot be understood in depth without the help of the theory of operators. Contemporary examples are: the theory of functions of a real variable, integral equations, the calculus of variations, etc.
This theory, therefore, well deserves, for its aesthetic value as much as for the scope of its arguments (even ignoring its numerous applications) the interest that it is attracting from more and more mathematicians. The opinion of J. Hadamard, who considers the theory of operators one of the most powerful methods of contemporary research in mathematics, should come as no surprise. |
