Kernel of Linear Transformation is Orthocomplement of Image of Adjoint

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

Let $\map \BB {\HH, \KK}$ denote the set of bounded linear transformations from $\HH$ to $\KK$.

Let $A \in \map \BB {\HH, \KK}$ 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 $x \in \HH$ be arbitrary.

Then:

Hence by definition of set equality:


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