# Definition:Closed Linear Span

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## 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 subspaces $M$ 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

The closed linear span of $A$ is often written $\map {\operatorname {clin} } A$.

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

- Definition:Linear Span, which justifies the nomenclature by definition $(3)$.

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

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality: Exercise $4$