Banach algebra

A Banach algebra, in functional analysis, is an associative algebra over the real or complex numbers which at the same time is also a Banach space. The algebra multiplication and the Banach space norm are required to be related by the following inequality:

||xy|| ≤ ||x|| ||y||     for all x and y.

This insures that the multiplication operation is continuous.

Important examples of Banach algebras are the algebras of all linear continuous operators on a Banach space (with functional composition as multiplication), the algebras of bounded real- or complex-valued functions defined on some set (with pointwise multiplication) and the algebras of continuous real- or complex-valued functions on some compact space. Every C-star-algebra is a Banach algebra.

Several elementary functions which are defined via [power series]? may be defined on any Banach algebra; examples include the exponential function and the trigonometric functions.