Concept

# Banach algebra

Summary
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy |x , y| \ \leq |x| , |y| \quad \text{ for all } x, y \in A. This ensures that the multiplication operation is continuous. A Banach algebra is called unital if it has an identity element for the multiplication whose norm is 1, and commutative if its multiplication is commutative. Any Banach algebra A (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra A_e so as to form a closed ideal of A_e. Often one assumes a priori that the algebra under consideration is unital: for one can develo
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