\begin{definition}[Definition:Bounded Sesquilinear Form]
Let $\HH, \KK$ be Hilbert spaces over $\Bbb F \in \set {\R, \C}$.
Let $u: \HH \times \KK \to \Bbb F$ be a sesquilinear form.
Then $u$ is a '''bounded sesquilinear form''', or to be '''bounded''', {{iff}}:
:$\exists M \in \R: \forall h \in \HH, k \in \KK: \size {\map u {h, k} } \le M \norm h_\HH \norm k_\KK$
A constant $M$ satisfying the above is called a '''bound''' for $u$.
\end{definition}