Definition:Closed Linear Span

Definition
Let $H$ be a Hilbert space over $\Bbb F \in \left\{{\R, \C}\right\}$, and let $A \subseteq H$ be a subset.

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


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

These definitions are equivalent, as proved in Equivalence of Definitions of Closed Linear Span.

When $\Bbb A$ is a collection of subsets of $H$, the notation $\vee \Bbb A$ is often used for $\displaystyle \vee \left({\bigcup \Bbb A}\right)$.

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

Also see

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


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