Dense Subspaces: Intersection
Hilbert Space: $\mathcal{H}$
Dense Subspaces: $$\mathcal{D},\mathcal{D}'\leq\mathcal{H}:\quad\overline{\mathcal{D}},\overline{\mathcal{D}'}=\mathcal{H}\not\Rightarrow\mathcal{D}\cap\mathcal{D}'\neq\{0\}$$ (Counterexample?)
Let $\mathcal{H}=L^2[0,1]$. Let $\mathcal{D}$ be the subspace of polynomials on $[0,1]$. Let $\mathcal{D}'$ be the subspace generated by $\{ \sin(n\pi x) \}_{n=1}^{\infty}$. Both are dense, and they have nothing in common except the $0$ vector.