Theorem bdopf

Description: A bounded linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 2-Feb-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	bdopf	$$⊢T\in \mathrm{BndLinOp}\to T:\mathscr{H}⟶\mathscr{H}$$

Proof

Step	Hyp	Ref	Expression
1		bdopln	$$⊢T\in \mathrm{BndLinOp}\to T\in \mathrm{LinOp}$$
2		lnopf	$$⊢T\in \mathrm{LinOp}\to T:\mathscr{H}⟶\mathscr{H}$$
3	1 2	syl	$$⊢T\in \mathrm{BndLinOp}\to T:\mathscr{H}⟶\mathscr{H}$$