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Banach algebra
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{{Short description|Particular kind of algebraic structure}} In [[mathematics]], especially [[functional analysis]], a '''Banach algebra''', named after [[Stefan Banach]], is an [[associative algebra]] <math>A</math> over the [[real number|real]] or [[complex number|complex]] numbers (or over a [[nonarchimedean field|non-Archimedean]] complete [[Norm (mathematics)|normed field]]) that at the same time is also a [[Banach space]], that is, a [[normed space]] that is [[complete metric space|complete]] in the [[metric (mathematics)|metric]] induced by the norm. The norm is required to satisfy <math display=block>\|x \, y\| \ \leq \|x\| \, \|y\| \quad \text{ for all } x, y \in A.</math> This ensures that the multiplication operation is [[continuous function (topology)|continuous]] with respect to the [[metric topology]]. A Banach algebra is called ''unital'' if it has an [[identity element]] for the multiplication whose norm is <math>1,</math> and ''commutative'' if its multiplication is [[commutative]]. Any Banach algebra <math>A</math> (whether it is unital or not) can be embedded [[isometry|isometrically]] into a unital Banach algebra <math>A_e</math> so as to form a [[closed set|closed]] [[ideal (algebra)|ideal]] of <math>A_e</math>. Often one assumes ''a priori'' that the algebra under consideration is unital because one can develop much of the theory by considering <math>A_e</math> 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 function]]s 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 (functional analysis)|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-adic number|<math>p</math>-adic number]]s. This is part of [[p-adic analysis|<math>p</math>-adic analysis]].
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