# Examples of Modules - For Normed Algebras X and Left X-Modules M, M* is a Banach Right X-Module

Let $X$ be a normed algebra and let $M$ be a normed left $X$-module. Let $M^*$ be the dual space of $M$ (the space of all bounded linear functionals on $M$). Then $M^*$ is a right $X$-module with the following module multiplication from $X \times M^* \to M^*$ as follows.

For each $x \in X$ and each $f \in M^*$ we define $(fx)$ to be the functional defined for all $m \in M$ by:

(1)Note that $(fx)$ is well-defined. Since $M$ is a left $X$-module we have that the multiplication $xm$ is defined and contained in $M$ (which is significant since $f \in M^*$).

**Showing that for every $f \in M^*$ and every $x \in X$ that $(fx) \in M^*$:**

For linearity, let $m_1, m_2 \in M$. Then:

(2)Let $m \in M$ and $a \in \mathbf{F}$. Then:

(3)And for boundedness we have that for every $m \in M$:

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

**Checking the axioms $LM1$, $LM2$, and $LM3$:**

For each fixed $x \in X$, we have that for all $f_1, f_2 \in M^*$ that:

(5)And thus for each $x \in X$, multiplication $(x, f) \to (fx)$ is linear and so $LM1$ is satisfied.

Similarly, for each fixed $f \in M^*$ we have that for all $x_1, x_2 \in X$ that:

(6)And thus for each $f \in M^*$, multiplication $(x, f) \to (fx)$ is linear and so $LM2$ is satisfied.

Lastly, for all $x_1, x_2 \in X$ and all $f \in M^*$ we have that:

(7)So $LM3$ is satisfied and so $M^*$ is an right $X$-module.

**Showing that there exists a $K > 0$ such that $\| (fx) \| \leq K \| f \| \| x \|$ for all $f \in M^*$ and all $x \in X$:**

It is furthermore a normed right $X$-module since for all $x \in X$ and all $f \in M^*$ we have that:

(8)Note that since $f$ is a bounded linear functional we have that $|f(xm)| \leq \| f \| \| xm \|_M \leq \| f \|_{\mathrm{op}} \| x \|_X \| m \|_M$, and thus:

(9)So $M^*$ is a normed right $X$-module and furthermore, since $M^* = \mathcal B(M, \mathbf{F})$ is always a Banach space we have that $M^*$ is a Banach right $X$-module.

It can similarly be shown that if $X$ is a normed algebra and $M$ is a normed right $X$-module then $M^*$ is a Banach left $X$-module.