Normed Ring

an important concept of functional analysis that greatly extends the range of applications of functional analysis. The elements of a normed ring are at the same time the points of a certain geometric structure—a complete normed vector space—and the elements of a certain algebraic structure—a ring—in which multiplication by numbers is defined, with the algebraic operations being continuous with respect to the norm. Some examples of normed rings are the ring C of all continuous functions on the closed interval [0,1] with the usual algebraic operations and with norm

the ring of matrices of order n, the ring of bounded operators on Hilbert space, and the ring L₁ of all functions that are absolutely integrable on the line with multiplication defined as convolution

The most highly developed of these theories is that of commutative normed rings, that is, normed rings in which multiplication is commutative: xy = yx. This theory was developed by I. M. Gel’fand.

Normed rings are also called Banach algebras.