\begin{definition}[Definition:Closed Linear Span]
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.
\end{definition}