Theorem spansnsh

Description: The span of a Hilbert space singleton is a subspace. (Contributed by NM, 17-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	spansnsh	$$⊢A\in \mathscr{H}\to \mathrm{span}\left(\left\{A\right\}\right)\in {\mathbf{S}}_{\mathscr{H}}$$

Proof

Step	Hyp	Ref	Expression
1		spansnch	$$⊢A\in \mathscr{H}\to \mathrm{span}\left(\left\{A\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
2		chsh	$$⊢\mathrm{span}\left(\left\{A\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}\to \mathrm{span}\left(\left\{A\right\}\right)\in {\mathbf{S}}_{\mathscr{H}}$$
3	1 2	syl	$$⊢A\in \mathscr{H}\to \mathrm{span}\left(\left\{A\right\}\right)\in {\mathbf{S}}_{\mathscr{H}}$$