Adjoint is Involutive

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

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

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

Define:


 * $A^{\ast \ast} = \paren {A^\ast}^\ast$

where $A^*$ denotes the adjoint of $A$.

Then $\ast$ is an involutive operation in the sense that:
 * $A^{\ast \ast} = A$

Proof
Let $x \in \HH$ and $y \in \KK$.

By the definition of the adjoint, we have:


 * $\innerprod {A y} x_\HH = \innerprod y {A^\ast x}_\KK$

Taking complex conjugates, we have:


 * $\overline {\innerprod {A y} x_\HH} = \overline {\innerprod y {A^\ast x}_\KK}$

Using the conjugate symmetry of the inner product, we have:


 * $\innerprod x {A y}_\HH = \innerprod {A^\ast x} y_\KK$

So by the definition of the adjoint:


 * $\innerprod x {A y}_\HH = \innerprod x {\paren {A^\ast}^\ast y}_\KK$

From the uniqueness of the adjoint, we obtain:


 * $\paren {A^\ast}^\ast = A$