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 $\lor A$, is defined in the following ways:


 * $(1): \qquad \displaystyle \lor A = \bigcap \Bbb M$, where $\Bbb M$ consists of all closed linear subspaces $M$ of $H$ with $A \subseteq M$
 * $(2): \qquad \displaystyle \lor A$ is the smallest closed linear subspace $M$ of $H$ with $A \subseteq M$
 * $(3): \qquad \displaystyle \lor A = \operatorname{cl} \left({\left\{{\sum_{k=1}^n \alpha_k f_k: n \in \N_{\ge 1}, \alpha_i \in \Bbb F, f_i \in A}\right\}}\right)$, where $\operatorname{cl}$ denotes closure

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

Furthermore, the closed linear span is also characterised by the identity $\lor A = (A^\perp)^\perp$, as proved in Double Orthocomplement is Closed Linear Span.