Banach algebra - definition. What is Banach algebra
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%ما هو (من)٪ 1 - تعريف

PARTICULAR KIND OF ALGEBRAIC STRUCTURE
Banach algebras; Spectrum of a commutative Banach algebra; Commutative Banach algebra; B-algebra; Banach *-algebra; Banach ring; Structure space; Spectral mapping theorem; Character space; Unital Banach algebra; Algebra norm; Involutive Banach algebra; Spectrum of an abelian Banach algebra

Banach algebra         
<mathematics> An algebra in which the vector space is a Banach space. (1997-02-25)
Banach algebra         
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
Amenable Banach algebra         
Amenable banach algebra; Amenable algebra
In mathematics, specifically in functional analysis, a Banach algebra, A, is amenable if all bounded derivations from A into dual Banach A-bimodules are inner (that is of the form a\mapsto a.x-x.

ويكيبيديا

Banach algebra

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle 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

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 , {\displaystyle 1,} and commutative if its multiplication is commutative. Any Banach algebra A {\displaystyle A} (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra A e {\displaystyle A_{e}} so as to form a closed ideal of A e {\displaystyle A_{e}} . Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering A e {\displaystyle A_{e}} and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the trigonometric functions in a Banach algebra without identity.

The theory of real Banach algebras can be very different from the theory of complex Banach algebras. For example, the spectrum of an element of a nontrivial complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements.

Banach algebras can also be defined over fields of p {\displaystyle p} -adic numbers. This is part of p {\displaystyle p} -adic analysis.