Standard Subspaces

Definition (Cyclic, Separating, Standard Subspace)

A closed real linear subspace $H$ of a complex Hilbert space $\mathcal{H}$ is called

• cyclic, if $H+iH$ is dense in $\mathcal{H}$,
• separating, if $H\cap iH=\{0\}$, and
• standard, if $H$ is cyclic and separating.