functional analysismathematics

In the 1920s and ’30s a number of apparently different areas of analysis all came together in a single generalization—rather, two generalizations, one more general than the other. These were the notions of a Hilbert space and a Banach space, named after the German mathematician David Hilbert and the Polish mathematician Stefan Banach, respectively. Together they laid the foundations for...

Hadamard’s Leçons sur le calcul des variations (1910; “Lessons on the Calculus of Variations”) helped to lay the foundations of the modern theory of functional analysis, in connection with which he introduced the term functional. Part of his work in determinants is important in the theory of integral equations.

Hilbert’s work in integral equations in about 1909 led directly to 20th-century research in functional analysis (the branch of mathematics in which functions are studied collectively). His work also established the basis for his work on infinite-dimensional space, later called Hilbert space, a concept that is useful in mathematical analysis and quantum mechanics. Making use of his results on...

In analysis, the discovery of Hilbert space ushered in functional analysis, a new field in which mathematicians study the properties of quite general linear spaces. Among these spaces are the complete inner product spaces, which now are called Hilbert spaces, a designation first used in 1929 by the Hungarian-American mathematician John von Neumann to describe these spaces in an abstract...

...them; useful examples include certain spaces of sequences and certain spaces of functions. Operators defined on these spaces are also of great interest; their study forms part of the field of functional analysis.

Many of Riesz’s fundamental findings in functional analysis were incorporated with those of Stefan Banach of Poland. The Riesz-Fischer theorem of 1907, concerning the equivalence of the Hilbert space of sequences of convergent sums of squares with the space of functions of summable squares, formed the mathematical basis for demonstrating the equivalence of matrix mechanics and wave mechanics, a...

French mathematician who was awarded the Fields Medal in 1950 for his work in functional analysis.

...Enrico Betti while the former attended the University of Pisa (1878–82). Volterra was appointed professor of rational mechanics at Pisa in 1883, the year he began devising a general theory of functionals (functions that depend on a continuous set of values of another function). This concept led to the development of new fields of analysis, including important applications to the solution...