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 $K^\perp = \left({0}\right)$, where $K^\perp$ is the orthocomplement of $K$, and $\left({0}\right)$ 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.