---
title: Standard Subspaces
parent: Tomita Takesaki Theory
nav_order: 1
# cspell:words
---

# {{ page.title }}

{: .definition-title }
> Definition (Cyclic, Separating, Standard Subspace)
>
> A closed real linear subspace $H$ of a complex Hilbert space $\hilb{H}$ is called
> * *cyclic*, if $H+iH$ is dense in $\hilb{H}$,
> * *separating*, if $H \cap iH = \braces{0}$, and
> * *standard*, if $H$ is cyclic and separating.

**Proof:**
{{ site.qed }}