Equivalence of Definitions of Closed Linear Span

Theorem

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

The three definitions of the closed linear span of $A$, viz.:

$(1): \qquad \displaystyle \vee A = \bigcap \Bbb M$, 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 $A \subseteq M$

$(3): \qquad \displaystyle \vee A = \operatorname{cl} \left({\left\{{\sum_{k=1}^n \alpha_k f_k: n \in \N_{\ge 1}, \alpha_i \in \Bbb F, f_i \in A}\right\}}\right)$, where $\operatorname{cl}$ denotes closure

are equivalent.

Sources

• John B. Conway: A Course in Functional Analysis (1990) $I.2 \text{ Exercise } 4$