A Banach algebra, in functional analysis, is an associative algebra 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:
- ||xy|| ≤ ||x|| ||y|| for all x and y.
- (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.
Important examples of Banach algebras are the algebras of all linear continuous operators on a Banach space (with functional composition as multiplication), the algebras of bounded real- or complex-valued functions defined on some set (with pointwise multiplication) and the algebras of continuous real- or complex-valued functions on some compact space. Every C-star-algebra is a Banach algebra.