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 ∈ ℋ
Assertion	spansnchi	⊢ ( span ‘ { 𝐴 } ) ∈ Cℋ

Proof

Step	Hyp	Ref	Expression
1		spansnch.1	⊢ 𝐴 ∈ ℋ
2		spansnch	⊢ ( 𝐴 ∈ ℋ → ( span ‘ { 𝐴 } ) ∈ Cℋ )
3	1 2	ax-mp	⊢ ( span ‘ { 𝐴 } ) ∈ Cℋ