# Examples of Modules - The Dual Banach Right Module of a Normed Left A-Module M

Let $\mathfrak{A}$ be a normed algebra and let $M$ be a normed left $\mathfrak{A}$-module. Let $M^*$ be the dual space of $M$ (the space of all bounded linear functionals on $M$). Then $M^*$ is a Banach right $\mathfrak{A}$-module called the **Dual Banach Right Module of $M$**, with the following module multiplication from $\mathfrak{A} \times M^* \to M^*$ as follows.

For each $a \in \mathfrak{A}$ and each $f \in M^*$ we define $(fa)$ to be the functional defined for all $m \in M$ by:

(1)Note that $(fa)$ is well-defined. Since $M$ is a left $\mathfrak{A}$-module we have that the multiplication $am$ is defined and contained in $M$ (which is significant since $f \in M^*$).

It is first important to check that each $fa$ is also in $M^*$.

First $fa$ is linear since for all $m_1, m_2 \in M$ we have that:

(2)And for all $m \in M$ and $\alpha \in \mathbf{F}$:

(3)Lastly, each $fa$ is bounded since for all $m \in M$:

(4)So $(fa)$ is bounded with $\| fa \| \leq \| f \| \| a \|$. So $(fa) \in M^*$.

We now check that $M^*$ is a Banach right $\mathfrak{A}$-module:

**1.**For each fixed $a \in \mathfrak{A}$, we have that for all $f_1, f_2 \in M^*$ that:

- And thus for each $a \in \mathfrak{A}$, multiplication $(a, f) \to (fa)$ is linear and so $RM1$ is satisfied.

**2.**Similarly, for each fixed $f \in M^*$ we have that for all $a_1, a_2 \in \mathfrak{A}$ that:

- And thus for each $f \in M^*$, multiplication $(a, f) \to (fa)$ is linear and so $RM2$ is satisfied.

**3.**Lastly, for all $a_1, a_2 \in \mathfrak{A}$ and all $f \in M^*$ we have that:

- So $RM3$ is satisfied and so $M^*$ is an right $\mathfrak{A}$-module.

**4.**For all $a \in \mathfrak{A}$ and all $f \in M^*$ we have that:

- Note that since $f$ is a bounded linear functional we have that $|f(am)| \leq \| f \| \| am \|_M \leq \| f \|_{\mathrm{op}} \| a \|_{\mathfrak{A}} \| m \|_M$, and thus:

So $M^*$ is a Banach right $\mathfrak{A}$-module.

It can similarly be shown that if $\mathfrak{A}$ is a normed algebra and $M$ is a normed right $\mathfrak{A}$-module then $M^*$ is a Banach left $\mathfrak{A}$-module.