Existence and Uniqueness of Adjoint

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

Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ and $\struct {\KK, \innerprod \cdot \cdot_\KK}$ be Hilbert spaces over $\mathbb F$.

Let $A: \HH \to \KK$ be a bounded linear transformation.

Then there exists a unique bounded linear transformation $B: \KK \to \HH$ such that:


 * $\innerprod {\map A x} y_\KK = \innerprod x {\map B y}_\HH$

for all $x \in \HH$ and $y \in \KK$.

That is:


 * each bounded linear transformation between Hilbert spaces has a unique adjoint.

Proof
We first show that such a unique mapping $B$ exists, without first insisting on a bounded linear transformation.

Lemma 1
We now show that $B$ is a linear transformation.

Lemma 2
Finally, we show that $B$ is bounded.

Lemma 3
So $B$ is the unique bounded linear transformation such that:


 * $\innerprod {\map A x} y_\KK = \innerprod x {\map B y}_\HH$

for all $x \in \HH$ and $y \in \KK$.