Definition:Closed Linear Span

Definition
Let $H$ be a Hilbert space, and let $A \subseteq H$ be a subset.

Then the closed linear span of $A$, denoted $\vee A$, is defined in the following ways:


 * $(1): \quad \ds \vee A = \bigcap \Bbb M$, where $\Bbb M$ consists of all closed linear subspace $M$ of $H$ with $A \subseteq M$
 * $(2): \quad \vee A$ is the smallest closed linear subspace $M$ of $H$ with $A \subseteq M$
 * $(3): \quad \vee A = \map \cl {\map \span A}$, where $\cl$ denotes closure, and $\span$ denotes linear span.

Also denoted as
When $\Bbb A$ is a collection of subsets of $H$, the notation $\vee \Bbb A$ is often used for $\ds \map \vee {\bigcup \Bbb A}$.

When $\Bbb A = \family {A_i: i \in I}$ is an $I$-indexed collection of subsets of $H$, also $\vee_i A_i$ may be encountered.

Also see

 * Equivalence of Definitions of Closed Linear Span.


 * Linear Span, which justifies the nomenclature by definition $(3)$.


 * Double Orthocomplement is Closed Linear Span, for another characterisation of the closed linear span.