Definition:Bounded Sesquilinear Form

Definition
Let $H, K$ be Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$.

Let $u: H \times K \to \Bbb F$ be a sesquilinear form.

Then $u$ is said to be a bounded sesquilinear form, or to be bounded, iff:


 * $\exists M \in \R: \forall h \in H, k \in K: \left\vert{u \left({h, k}\right)}\right\vert \le M \left\Vert{h}\right\Vert_H \left\Vert{k}\right\Vert_K$

A constant $M$ satisfying the above is called a bound for $u$.

Also see

 * Bounded Linear Transformation Induces Bounded Sesquilinear Form, which establishes a class of examples of sesquilinear forms.
 * Classification of Bounded Sesquilinear Forms, which states that the above class of examples is complete; i.e., there are no other sesquilinear forms.