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- Banach_algebra abstract "In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers that at the same time is also a Banach space, i.e. normed and complete. The algebra multiplication and the Banach space norm are required to be related by the following inequality:(i.e., the norm of the product is less than or equal to the product of the norms). This ensures that the multiplication operation is continuous. This property is found in the real and complex numbers; for instance, |-6×5| ≤ |-6|×|5|.If in the above we relax Banach space to normed space the analogous structure is called a normed algebra.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 (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra so as to form a closed ideal of . Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering 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-adic numbers. This is part of p-adic analysis.".
- Banach_algebra wikiPageID "4665".
- Banach_algebra wikiPageRevisionID "606053990".
- Banach_algebra hasPhotoCollection Banach_algebra.
- Banach_algebra subject Category:Banach_algebras.
- Banach_algebra subject Category:Fourier_analysis.
- Banach_algebra subject Category:Science_and_technology_in_Poland.
- Banach_algebra type Abstraction100002137.
- Banach_algebra type Algebra106012726.
- Banach_algebra type BanachAlgebras.
- Banach_algebra type Cognition100023271.
- Banach_algebra type Content105809192.
- Banach_algebra type Discipline105996646.
- Banach_algebra type KnowledgeDomain105999266.
- Banach_algebra type Mathematics106000644.
- Banach_algebra type PsychologicalFeature100023100.
- Banach_algebra type PureMathematics106003682.
- Banach_algebra type Science105999797.
- Banach_algebra comment "In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers that at the same time is also a Banach space, i.e. normed and complete. The algebra multiplication and the Banach space norm are required to be related by the following inequality:(i.e., the norm of the product is less than or equal to the product of the norms). This ensures that the multiplication operation is continuous.".
- Banach_algebra label "Algebra Banacha".
- Banach_algebra label "Algebra di Banach".
- Banach_algebra label "Algebras de Banach".
- Banach_algebra label "Algèbre de Banach".
- Banach_algebra label "Banach algebra".
- Banach_algebra label "Banach-algebra".
- Banach_algebra label "Banachalgebra".
- Banach_algebra label "Álgebra de Banach".
- Banach_algebra label "Банахова алгебра".
- Banach_algebra label "バナッハ環".
- Banach_algebra sameAs Banachova_algebra.
- Banach_algebra sameAs Banachalgebra.
- Banach_algebra sameAs Algebras_de_Banach.
- Banach_algebra sameAs Algèbre_de_Banach.
- Banach_algebra sameAs Algebra_di_Banach.
- Banach_algebra sameAs バナッハ環.
- Banach_algebra sameAs 바나흐_대수.
- Banach_algebra sameAs Banach-algebra.
- Banach_algebra sameAs Algebra_Banacha.
- Banach_algebra sameAs Álgebra_de_Banach.
- Banach_algebra sameAs m.01h6c.
- Banach_algebra sameAs Q806066.
- Banach_algebra sameAs Q806066.
- Banach_algebra sameAs Banach_algebra.
- Banach_algebra wasDerivedFrom Banach_algebra?oldid=606053990.
- Banach_algebra isPrimaryTopicOf Banach_algebra.