Examples of Modules - For Banach Left A-Modules M, l1(M) is a Banach Left A-Module
Let $\mathfrak{A}$ be a Banach algebra and again let $\{ M_{\alpha} : \alpha \in \Lambda \}$ be a collection of Banach left $\mathfrak{A}$-modules such that there exists a $K > 0$ such that:
(1)for all $a \in \mathfrak{A}$ and for all $m \in M_{\alpha}$, $\alpha \in \Lambda$. Let:
(2)We equip this space with the operations of pointwise function addition and scalar multiplication to make it a linear space.
We define a norm on $l^1(M_{\alpha} : \alpha \in \Lambda)$ to make it a normed linear space. This norm is defined for all $f \in l^1(M_{\alpha} : \alpha \in \Lambda)$ by:
(3)$l^1(M_{\alpha} : \alpha \in \Lambda)$ becomes a Banach space with this norm. In fact, this space becomes a Banach left $\mathfrak{A}$-module with the module multiplication defined for all $f \in l^1(M_{\alpha} : \alpha \in \Lambda)$ and all $a \in \mathfrak{A}$ by:
(4)(Again observe that $f(\alpha) \in M_{\alpha}$. Since $M_{\alpha}$ is a Banach left $\mathfrak{A}$-module we have that $af(\alpha)$ is defined and furthermore that $af(\alpha) \in M_{\alpha}$).
Showing that for all $a \in \mathfrak{A}$ and for all $f \in l^1(M_{\alpha} : \alpha \in \Lambda)$ that $af \in l^1(M_{\alpha} : \alpha \in \Lambda)$:
We first need to show that for all $f \in l^1(M_{\alpha} : \alpha \in \Lambda)$ and for all $a \in \mathfrak{A}$ we have that $af \in l^1(M_{\alpha} : \alpha \in \Lambda)$. We have that:
(5)So $af \in l^1(M_{\alpha} : \alpha \in \Lambda)$.
Lastly we have that:
(6)So $l^1(M_{\alpha} : \alpha \in \Lambda)$ is a Banach left $\mathfrak{A}$-module.
Again by taking $\Lambda = \mathbb{N}$ and $M_n = M$ for all $n \in \mathbb{N}$ we obtain $l^1(M) := l^1(M_n : n \in \mathbb{N})$ as a Banach left $\mathfrak{A}$-module.