Bounded Linear Transformation Induces Bounded Sesquilinear Form

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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.


Proof


Also see


Sources