Adjoint is Involutive

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

Let $\HH$ and $\KK$ be Hilbert spaces.

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

Define:


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

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

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

Proof
Let $\innerprod \cdot \cdot_\HH$ and $\innerprod \cdot \cdot_\KK$ be inner products on $\HH$ and $\KK$ respectively.

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

By the definition of the adjoint, we have:


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

Taking complex conjugates, we have:


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

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


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

So by the definition of the adjoint, we have:


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