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

• Equivalence of Definitions of Closed Linear Span

• 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 $4$