Kernel of Linear Transformation is Orthocomplement of Image of Adjoint

Theorem
Let $H, K$ be Hilbert spaces.

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

Then $\operatorname{ker} A = \left({\operatorname{ran} A^*}\right)^\perp$, where:


 * $A^*$ denotes the adjoint of $A$
 * $\operatorname{ker} A$ is the kernel of $A$
 * $\operatorname{ran} A^*$ is the range of $A^*$
 * $\perp$ signifies orthocomplementation

Proof
Let $x \in H$ be arbitrary. Then:

Hence by definition of set equality:


 * $\operatorname{ker} A = \left({\operatorname{ran} A^*}\right)^\perp$