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...