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+---
+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 }}