---
title: Standard Subspaces
parent: Tomita Takesaki Theory
nav_order: 1
---

# {{ page.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.
{% enddefinition %}