# Classification of Bounded Sesquilinear Forms

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## Contents

## Theorem

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

Let $u: H \times K \to \Bbb F$ be a bounded sesquilinear form with bound $M$.

Then there exist unique bounded linear transformations $A \in B \left({H, K}\right), B \in B \left({K, H}\right)$ such that:

- $\forall h \in H, k \in K: u \left({h, k}\right) = \left\langle{Ah, k}\right\rangle_K = \left\langle{h, Bk}\right\rangle_H$

Furthermore, $\left\Vert{A}\right\Vert, \left\Vert{B}\right\Vert \le M$.

## Proof

## Also see

- Bounded Linear Transformation Induces Bounded Sesquilinear Form, which shows that considered inner products are in fact bounded sesquilinear forms.

- The notion of the adjoint of a bounded linear transformation, which relies on this theorem for being well-defined.

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $II.2.2$