# Definition:Bounded Sesquilinear Form

## Definition

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, if and only if:

$\exists M \in \R: \forall h \in \HH, k \in \KK: \size {\map u {h, k} } \le M \norm h_\HH \norm k_\KK$

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

## Linguistic Note

Sesqui means one-and-a-half, deriving from the Latin for and also a half.

This gives rise to the term sesquilinear if one regards conjugate linearity as being almost or half linearity.