Theorem hlobn

Description: Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007) (New usage is discouraged.)

	Ref	Expression
Assertion	hlobn	$$⊢U\in {CHil}_{\mathrm{OLD}}\to U\in \mathrm{CBan}$$

Proof

Step	Hyp	Ref	Expression
1		ishlo	$$⊢U\in {CHil}_{\mathrm{OLD}}\leftrightarrow \left(U\in \mathrm{CBan}\wedge U\in {CPreHil}_{\mathrm{OLD}}\right)$$
2	1	simplbi	$$⊢U\in {CHil}_{\mathrm{OLD}}\to U\in \mathrm{CBan}$$