Adjoint is Involutive

Theorem
Let $H, K$ be Hilbert spaces.

Let $A \in B \left({H, K}\right)$ be a bounded linear transformation.

Then $A^{**} := \left({A^*}\right)^* = A$.

Proof
Let $h \in H, k \in K$. Then:

Thus, by Existence and Uniqueness of Adjoint, $A^{**} = A$.