Linear Subspace Dense iff Zero Orthocomplement

Theorem
Let $H$ be a Hilbert space.

Let $K$ be a linear subspace of $H$.

Then $K$ is everywhere dense $K^\perp = \paren 0$, where $K^\perp$ is the orthocomplement of $K$, and $\paren 0$ denotes the zero subspace.

Sufficient Condition
Assume that $K$ is everywhere dense and $x \in K^\perp$.

Then:

Necessary Condition
Assume $K^\perp = 0$.

Then by Double Orthocomplement is Closed Linear Span:
 * $\vee K = \paren 0^\perp = H$

Hence $K$ is everywhere dense.