Classification of Bounded Sesquilinear Forms

Theorem

Let $\mathbb F \in \set {\R, \C}$.

Let $\HH$ a be Hilbert space over $\mathbb F$ with inner product ${\innerprod \cdot \cdot}_\HH$.

Let $\KK$ a be Hilbert space over $\mathbb F$ with inner product ${\innerprod \cdot \cdot}_\KK$.

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

Then there exist unique bounded linear transformations $A : \HH \to \KK$ and $B : \KK \to \HH$ such that:

$\forall h \in \HH, k \in \KK: \map u {h, k} = \innerprod {A h} k_\KK = \innerprod h {B k}_\HH$

Furthermore:

$\norm A \le M$

and:

$\norm B \le M$

where $\norm \cdot$ denotes the norm of a bounded linear transformation.

Proof

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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 (2nd ed.) ... (previous) ... (next) $\text {II}.2.2$