# Definition:Bounded Sesquilinear Form

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## 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) $\S \text {II}.2$