In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A 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:
(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 continu...
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In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A 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:
(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.
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 A (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra Ae so as to form a closed ideal of Ae. Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory...
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