Bounded Linear Transformation Induces Bounded Sesquilinear Form

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

Let $A \in B \left({H, K}\right), B \in B \left({K, H}\right)$ be bounded linear transformations.

Let $u, v: H \times K \to \Bbb F$ be defined by:


 * $u \left({h, k}\right) := \left\langle{Ah, k}\right\rangle_K$
 * $v \left({h, k}\right) := \left\langle{h, Bk}\right\rangle_H$

Then $u$ and $v$ are bounded sesquilinear forms.

Also see

 * Classification of Bounded Sesquilinear Forms, which states that all sesquilinear forms are of this type.