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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 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, and commutative if its multiplication is commutative.Any Banach algebra A {\displaystyle A} 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.