Kernel of Linear Transformation is Orthocomplement of Image of Adjoint

Theorem
Let $H, K$ be Hilbert spaces.

Let $A \in B \struct {H, K}$ be a bounded linear transformation.

Then $\ker A = \paren {\Img {A^*} }^\perp$, where:


 * $A^*$ denotes the adjoint of $A$
 * $\ker A$ is the kernel of $A$
 * $\Img {A^*}$ is the image of $A^*$
 * $\perp$ signifies orthocomplementation

Proof
Let $x \in H$ be arbitrary.

Then:

Hence by definition of set equality:


 * $\ker A = \paren {\Img {A^*} }^\perp$