Hadamard, Jacques Salomon (1865–1963)

French mathematician. He originated functional analysis, one of the most fertile branches of modern mathematics. He also made contributions to number theory and formulated the concept of a correctly posed problem.

Functions Hadamard's early work was on analytic functions; that is, functions that can be developed as power series that converge. He began to study the Riemann zeta function and in 1896 solved the problem of determining the number of prime numbers less than a given number x. Hadamard was able to demonstrate that this number was asymptotically (see asymptote) equal to x/log x, which was the most important single result ever obtained in number theory. Hadamard became interested in the ‘functions of lines’ – numerical functions that depend upon a curve or an ordinary function as their variable. By extending the theory of ordinary functions to the case where the variable, or variables, would no longer be a number, or numbers, Hadamard created a new branch of mathematics. This required a redefinition, or at least a new generalization, of concepts such as continuity, derivative, and differential.

Lacunary space By extension from this work, Hadamard came to investigate functions of a complex variable and to define a singularity as a point at which a function ceases to be regular. He showed that the existence of a set of singular points may be compatible with the continuity of a function, and named the region formed by such a set a ‘lacunary space’. The study of such spaces has occupied mathematicians ever since.

Correctly posed problem Since it is often helpful, or necessary, to find an approximate solution (in physics, for example), a correctly posed problem, according to Hadamard, is one for which a solution exists that is unique for given data, but which also depends continuously on the data. This is the case when the solution can be expressed as a set of convergent power series. The idea has been fundamental to the development of the theory of function spaces, functional analysis.

Life Hadamard was born in Versailles and studied at the Ecole Normale Supérieure in Paris. He was professor of mathematics at the Collège de France in Paris 1909–37. During the German occupation of France in World War II, he went into exile, returning in 1945.