Example

On the vector space (with respect to the field $\mathbf{R}$) of continuous real-valued functions on an interval $[a, b]$ of $\mathbf{R}$, the mapping $x \mapsto \int_a^b x(t) dt$ is a linear form.

Let $E$ be a left A-module and $E^*$ its dual; for every ordered pair of elements $x \in E, x^* \in E^*$, the element $x^*(x)$ of A is denoted by $\langle x, x^* \rangle$. Then the relations

hold for $x, y$ in $E, x^*, y^*$ in $E^*$ and $\alpha \in A$. The mapping $(x, x^*) \mapsto \langle x, x^* \rangle$ of $E \times E^*$ into $A$ is called the canonical bilinear form on $E \times E^*$ (the notion of bilinear form will be defined generally in IX, § 1). Every linear form $x^*$ on $E$ can be considered as the partial mapping $x \mapsto \langle x, x^* \rangle$ corresponding to the canonical bilinear form.

When $E$ is a right A-module, the value $x^*(x)$ of a linear form $x^* \in E^*$ at an element $x \in E$ is denoted by $\langle x^*, x \rangle$ and the formulae corresponding to (11) and (12) are written as

When A is commutative, either notation is permissible.