Theorem spansnchi

Description: The span of a singleton in Hilbert space is a closed subspace. (Contributed by NM, 3-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Hypothesis	spansnch.1	$$⊢A\in \mathscr{H}$$
Assertion	spansnchi	$$⊢\mathrm{span}\left(\left\{A\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}$$

Proof

Step	Hyp	Ref	Expression
1		spansnch.1	$$⊢A\in \mathscr{H}$$
2		spansnch	$$⊢A\in \mathscr{H}\to \mathrm{span}\left(\left\{A\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
3	1 2	ax-mp	$$⊢\mathrm{span}\left(\left\{A\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}$$