# Banach module

(left) over a Banach algebra $A$
A Banach space $X$ together with a continuous bilinear operator $m : A \times X \rightarrow X$ defining on $X$ the structure of a left module over $A$ in the algebraic sense. A right Banach module and a Banach bimodule over $A$ are defined in an analogous manner. A continuous homomorphism of two Banach modules is called a morphism. Examples of Banach modules over $A$ include a closed ideal in $A$ and a Banach algebra $B \supset A$. A Banach module over $A$ that can be represented as a direct factor of Banach modules $A_+ \hat\otimes E$, (where $A_+$ is $A$ with an added unit and $E$ is a Banach space and $m(a,b \otimes x) = ab \otimes x$) is called projective. Cf. Topological tensor product.