> [!definition] > > Let $E$ be a $\complex$-[[Vector Space|vector space]]. A mapping $\lambda: E^2 \to \complex$ is a **sesquilinear form** if for every $x_1, x_2, x, y_1, y_2, y \in E$, > 1. $\lambda(\alpha x_1 + x_2, y) = \alpha\lambda(x_1, y) + \lambda(x_2, y)$. > 2. $\lambda(x, \lambda y_1 + y_2) = \overline{\alpha}\lambda(x, y_1) + \lambda(x, y_2)$. > > If in addition, $\lambda(x, y) = \ol{\lambda(y, x)}$ for all $x, y \in E$, then $\lambda$ is a **Hermitian** form. > [!definition] > > Let $E$ be a [[Normed Vector Space|normed space]] and $\lambda: E^2 \to \complex$ be a sesquilinear form, then $E$ is **bounded** if there exists $C \ge 0$ such that > $ > \lambda(x, y) \le C\norm{x} \cdot \norm{y} \quad \forall x, y \in E > $ > and **coercive** if there exists $c \ge 0$ such that > $ > \lambda(x, x) \ge c\norm{x}^2 \quad \forall x, y \in E > $