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 $u$.

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; that is, there are no other sesquilinear forms.

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.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\S \text {II}.2$