P. 322 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32101-32200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hoaddcli 32101	Mapping of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
Theorem	hosubcli 32102	Mapping of difference of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ
Theorem	hoaddfni 32103	Functionality of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝑆 +op 𝑇) Fn ℋ
Theorem	hosubfni 32104	Functionality of difference of Hilbert space operators. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝑆 −op 𝑇) Fn ℋ
Theorem	hoaddcomi 32105	Commutativity of sum of Hilbert space operators. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝑆 +op 𝑇) = (𝑇 +op 𝑆)
Theorem	hosubcl 32106	Mapping of difference of Hilbert space operators. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 −op 𝑇): ℋ⟶ ℋ)
Theorem	hoaddcom 32107	Commutativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇) = (𝑇 +op 𝑆))
Theorem	hodsi 32108	Relationship between Hilbert space operator difference and sum. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)
⊢ 𝑅: ℋ⟶ ℋ    &   ⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ ((𝑅 −op 𝑆) = 𝑇 ↔ (𝑆 +op 𝑇) = 𝑅)
Theorem	hoaddassi 32109	Associativity of sum of Hilbert space operators. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
⊢ 𝑅: ℋ⟶ ℋ    &   ⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))
Theorem	hoadd12i 32110	Commutative/associative law for Hilbert space operator sum that swaps the first two terms. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
⊢ 𝑅: ℋ⟶ ℋ    &   ⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝑅 +op (𝑆 +op 𝑇)) = (𝑆 +op (𝑅 +op 𝑇))
Theorem	hoadd32i 32111	Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
⊢ 𝑅: ℋ⟶ ℋ    &   ⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆)
Theorem	hocadddiri 32112	Distributive law for Hilbert space operator sum. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
⊢ 𝑅: ℋ⟶ ℋ    &   ⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))
Theorem	hocsubdiri 32113	Distributive law for Hilbert space operator difference. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
⊢ 𝑅: ℋ⟶ ℋ    &   ⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ ((𝑅 −op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) −op (𝑆 ∘ 𝑇))
Theorem	ho2coi 32114	Double composition of Hilbert space operators. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)
⊢ 𝑅: ℋ⟶ ℋ    &   ⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝐴 ∈ ℋ → (((𝑅 ∘ 𝑆) ∘ 𝑇)‘𝐴) = (𝑅‘(𝑆‘(𝑇‘𝐴))))
Theorem	hoaddass 32115	Associativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇)))
Theorem	hoadd32 32116	Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆))
Theorem	hoadd4 32117	Rearrangement of 4 terms in a sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
⊢ (((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝑅 +op 𝑆) +op (𝑇 +op 𝑈)) = ((𝑅 +op 𝑇) +op (𝑆 +op 𝑈)))
Theorem	hocsubdir 32118	Distributive law for Hilbert space operator difference. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑅 −op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) −op (𝑆 ∘ 𝑇)))
Theorem	hoaddridi 32119	Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝑇 +op 0hop ) = 𝑇
Theorem	hodidi 32120	Difference of a Hilbert space operator from itself. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝑇 −op 𝑇) = 0hop
Theorem	ho0coi 32121	Composition of the zero operator and a Hilbert space operator. (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)
⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ ( 0hop ∘ 𝑇) = 0hop
Theorem	hoid1i 32122	Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝑇 ∘ Iop ) = 𝑇
Theorem	hoid1ri 32123	Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ ( Iop ∘ 𝑇) = 𝑇
Theorem	hoaddrid 32124	Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → (𝑇 +op 0hop ) = 𝑇)
Theorem	hodid 32125	Difference of a Hilbert space operator from itself. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → (𝑇 −op 𝑇) = 0hop )
Theorem	hon0 32126	A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅)
Theorem	hodseqi 32127	Subtraction and addition of equal Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝑆 +op (𝑇 −op 𝑆)) = 𝑇
Theorem	ho0subi 32128	Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝑆 −op 𝑇) = (𝑆 +op ( 0hop −op 𝑇))
Theorem	honegsubi 32129	Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝑆 +op (-1 ·op 𝑇)) = (𝑆 −op 𝑇)
Theorem	ho0sub 32130	Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 −op 𝑇) = (𝑆 +op ( 0hop −op 𝑇)))
Theorem	hosubid1 32131	The zero operator subtracted from a Hilbert space operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → (𝑇 −op 0hop ) = 𝑇)
Theorem	honegsub 32132	Relationship between Hilbert space operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 +op (-1 ·op 𝑈)) = (𝑇 −op 𝑈))
Theorem	homullid 32133	An operator equals its scalar product with one. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → (1 ·op 𝑇) = 𝑇)
Theorem	homco1 32134	Associative law for scalar product and composition of operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) ∘ 𝑈) = (𝐴 ·op (𝑇 ∘ 𝑈)))
Theorem	homulass 32135	Scalar product associative law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇)))
Theorem	hoadddi 32136	Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 +op 𝑈)) = ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)))
Theorem	hoadddir 32137	Scalar product reverse distributive law for Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·op 𝑇) = ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇)))
Theorem	homul12 32138	Swap first and second factors in a nested operator scalar product. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op (𝐵 ·op 𝑇)) = (𝐵 ·op (𝐴 ·op 𝑇)))
Theorem	honegneg 32139	Double negative of a Hilbert space operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → (-1 ·op (-1 ·op 𝑇)) = 𝑇)
Theorem	hosubneg 32140	Relationship between operator subtraction and negative. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 −op (-1 ·op 𝑈)) = (𝑇 +op 𝑈))
Theorem	hosubdi 32141	Scalar product distributive law for operator difference. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 −op 𝑈)) = ((𝐴 ·op 𝑇) −op (𝐴 ·op 𝑈)))
Theorem	honegdi 32142	Distribution of negative over addition. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇 +op 𝑈)) = ((-1 ·op 𝑇) +op (-1 ·op 𝑈)))
Theorem	honegsubdi 32143	Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇 −op 𝑈)) = ((-1 ·op 𝑇) +op 𝑈))
Theorem	honegsubdi2 32144	Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇 −op 𝑈)) = (𝑈 −op 𝑇))
Theorem	hosubsub2 32145	Law for double subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑆 −op (𝑇 −op 𝑈)) = (𝑆 +op (𝑈 −op 𝑇)))
Theorem	hosub4 32146	Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
⊢ (((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝑅 +op 𝑆) −op (𝑇 +op 𝑈)) = ((𝑅 −op 𝑇) +op (𝑆 −op 𝑈)))
Theorem	hosubadd4 32147	Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
⊢ (((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝑅 −op 𝑆) −op (𝑇 −op 𝑈)) = ((𝑅 +op 𝑈) −op (𝑆 +op 𝑇)))
Theorem	hoaddsubass 32148	Associative-type law for addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆 +op 𝑇) −op 𝑈) = (𝑆 +op (𝑇 −op 𝑈)))
Theorem	hoaddsub 32149	Law for operator addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆 +op 𝑇) −op 𝑈) = ((𝑆 −op 𝑈) +op 𝑇))
Theorem	hosubsub 32150	Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑆 −op (𝑇 −op 𝑈)) = ((𝑆 −op 𝑇) +op 𝑈))
Theorem	hosubsub4 32151	Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆 −op 𝑇) −op 𝑈) = (𝑆 −op (𝑇 +op 𝑈)))
Theorem	ho2times 32152	Two times a Hilbert space operator. (Contributed by NM, 26-Aug-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = (𝑇 +op 𝑇))
Theorem	hoaddsubassi 32153	Associativity of sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
⊢ 𝑅: ℋ⟶ ℋ    &   ⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ ((𝑅 +op 𝑆) −op 𝑇) = (𝑅 +op (𝑆 −op 𝑇))
Theorem	hoaddsubi 32154	Law for sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
⊢ 𝑅: ℋ⟶ ℋ    &   ⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ ((𝑅 +op 𝑆) −op 𝑇) = ((𝑅 −op 𝑇) +op 𝑆)
Theorem	hosd1i 32155	Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
⊢ 𝑇: ℋ⟶ ℋ    &   ⊢ 𝑈: ℋ⟶ ℋ    ⇒   ⊢ (𝑇 +op 𝑈) = (𝑇 −op ( 0hop −op 𝑈))
Theorem	hosd2i 32156	Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
⊢ 𝑇: ℋ⟶ ℋ    &   ⊢ 𝑈: ℋ⟶ ℋ    ⇒   ⊢ (𝑇 +op 𝑈) = (𝑇 −op ((𝑈 −op 𝑈) −op 𝑈))
Theorem	hopncani 32157	Hilbert space operator cancellation law. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
⊢ 𝑇: ℋ⟶ ℋ    &   ⊢ 𝑈: ℋ⟶ ℋ    ⇒   ⊢ ((𝑇 +op 𝑈) −op 𝑈) = 𝑇
Theorem	honpcani 32158	Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
⊢ 𝑇: ℋ⟶ ℋ    &   ⊢ 𝑈: ℋ⟶ ℋ    ⇒   ⊢ ((𝑇 −op 𝑈) +op 𝑈) = 𝑇
Theorem	hosubeq0i 32159	If the difference between two operators is zero, they are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
⊢ 𝑇: ℋ⟶ ℋ    &   ⊢ 𝑈: ℋ⟶ ℋ    ⇒   ⊢ ((𝑇 −op 𝑈) = 0hop ↔ 𝑇 = 𝑈)
Theorem	honpncani 32160	Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
⊢ 𝑅: ℋ⟶ ℋ    &   ⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ ((𝑅 −op 𝑆) +op (𝑆 −op 𝑇)) = (𝑅 −op 𝑇)
Theorem	ho01i 32161*	A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S8) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = 0 ↔ 𝑇 = 0hop )
Theorem	ho02i 32162*	A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S10) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = 0 ↔ 𝑇 = 0hop )
Theorem	hoeq1 32163*	A condition implying that two Hilbert space operators are equal. Lemma 3.2(S9) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑦) = ((𝑇‘𝑥) ·ih 𝑦) ↔ 𝑆 = 𝑇))
Theorem	hoeq2 32164*	A condition implying that two Hilbert space operators are equal. Lemma 3.2(S11) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆‘𝑦)) = (𝑥 ·ih (𝑇‘𝑦)) ↔ 𝑆 = 𝑇))
Theorem	adjmo 32165*	Every Hilbert space operator has at most one adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
⊢ ∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦))
Theorem	adjsym 32166*	Symmetry property of an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑆‘𝑥) ·ih 𝑦)))
Theorem	eigrei 32167	A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for an eigenvalue 𝐵 to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 21-Jan-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ))
Theorem	eigre 32168	A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for an eigenvalue 𝐵 to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ))
Theorem	eigposi 32169	A sufficient condition (first conjunct pair, that holds when 𝑇 is a positive operator) for an eigenvalue 𝐵 (second conjunct pair) to be nonnegative. Remark (ii) in [Hughes] p. 137. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))
Theorem	eigorthi 32170	A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for two eigenvectors 𝐴 and 𝐵 to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Jan-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐷 ∈ ℂ    ⇒   ⊢ ((((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0))
Theorem	eigorth 32171	A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for two eigenvectors 𝐴 and 𝐵 to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.)
⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) ∧ (((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))) → ((𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0))
20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms
Definition	df-nmop 32172*	Define the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
⊢ normop = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑧)))}, ℝ*, < ))
Definition	df-cnop 32173*	Define the set of continuous operators on Hilbert space. For every "epsilon" (𝑦) there is a "delta" (𝑧) such that... (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
⊢ ContOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑡‘𝑤) −ℎ (𝑡‘𝑥))) < 𝑦)}
Definition	df-lnop 32174*	Define the set of linear operators on Hilbert space. (See df-hosum 32063 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
⊢ LinOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑡‘𝑦)) +ℎ (𝑡‘𝑧))}
Definition	df-bdop 32175	Define the set of bounded linear Hilbert space operators. (See df-hosum 32063 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
⊢ BndLinOp = {𝑡 ∈ LinOp ∣ (normop‘𝑡) < +∞}
Definition	df-unop 32176*	Define the set of unitary operators on Hilbert space. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
⊢ UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦))}
Definition	df-hmop 32177*	Define the set of Hermitian operators on Hilbert space. Some books call these "symmetric operators" and others call them "self-adjoint operators", sometimes with slightly different technical meanings. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
⊢ HrmOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·ih 𝑦)}
20.6.5  Linear and continuous functionals and norms
Definition	df-nmfn 32178*	Define the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
⊢ normfn = (𝑡 ∈ (ℂ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ*, < ))
Definition	df-nlfn 32179	Define the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
⊢ null = (𝑡 ∈ (ℂ ↑m ℋ) ↦ (◡𝑡 “ {0}))
Definition	df-cnfn 32180*	Define the set of continuous functionals on Hilbert space. For every "epsilon" (𝑦) there is a "delta" (𝑧) such that... (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
⊢ ContFn = {𝑡 ∈ (ℂ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)}
Definition	df-lnfn 32181*	Define the set of linear functionals on Hilbert space. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
⊢ LinFn = {𝑡 ∈ (ℂ ↑m ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧))}
20.6.6  Adjoint
Definition	df-adjh 32182*	Define the adjoint of a Hilbert space operator (if it exists). The domain of adjℎ is the set of all adjoint operators. Definition of adjoint in [Kalmbach2] p. 8. Unlike Kalmbach (and most authors), we do not demand that the operator be linear, but instead show (in adjbdln 32416) that the adjoint exists for a bounded linear operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
⊢ adjℎ = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑢‘𝑦)))}
20.6.7  Dirac bra-ket notation
Definition	df-bra 32183*	Define the bra of a vector used by Dirac notation. Based on definition of bra in [Prugovecki] p. 186 (p. 180 in 1971 edition). In Dirac bra-ket notation, ⟨𝐴 ∣ 𝐵⟩ is a complex number equal to the inner product (𝐵 ·ih 𝐴). But physicists like to talk about the individual components ⟨𝐴 ∣ and ∣ 𝐵⟩, called bra and ket respectively. In order for their properties to make sense formally, we define the ket ∣ 𝐵⟩ as the vector 𝐵 itself, and the bra ⟨𝐴 ∣ as a functional from ℋ to ℂ. We represent the Dirac notation ⟨𝐴 ∣ 𝐵⟩ by ((bra‘𝐴)‘𝐵); see braval 32277. The reversal of the inner product arguments not only makes the bra-ket behavior consistent with physics literature (see comments under ax-his3 31417) but is also required in order for the associative law kbass2 32450 to work. Our definition of bra and the associated outer product df-kb 32184 differs from, but is equivalent to, a common approach in the literature that makes use of mappings to a dual space. Our approach eliminates the need to have a parallel development of this dual space and instead keeps everything in Hilbert space. For an extensive discussion about how our notation maps to the bra-ket notation in physics textbooks, see mmnotes.txt 32184, under the 17-May-2006 entry. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
⊢ bra = (𝑥 ∈ ℋ ↦ (𝑦 ∈ ℋ ↦ (𝑦 ·ih 𝑥)))
Definition	df-kb 32184*	Define a commuted bra and ket juxtaposition used by Dirac notation. In Dirac notation, ∣ 𝐴⟩⟨𝐵 ∣ is an operator known as the outer product of 𝐴 and 𝐵, which we represent by (𝐴 ketbra 𝐵). Based on Equation 8.1 of [Prugovecki] p. 376. This definition, combined with Definition df-bra 32183, allows any legal juxtaposition of bras and kets to make sense formally and also to obey the associative law when mapped back to Dirac notation. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
⊢ ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥)))
20.6.8  Positive operators
Definition	df-leop 32185*	Define positive operator ordering. Definition VI.1 of [Retherford] p. 49. Note that ( ℋ × 0ℋ) ≤op 𝑇 means that 𝑇 is a positive operator. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ ≤op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢 −op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥))}
20.6.9  Eigenvectors, eigenvalues, spectrum
Definition	df-eigvec 32186*	Define the eigenvector function. Theorem eleigveccl 32292 shows that eigvec‘𝑇, the set of eigenvectors of Hilbert space operator 𝑇, are Hilbert space vectors. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
⊢ eigvec = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·ℎ 𝑥)})
Definition	df-eigval 32187*	Define the eigenvalue function. The range of eigval‘𝑇 is the set of eigenvalues of Hilbert space operator 𝑇. Theorem eigvalcl 32294 shows that (eigval‘𝑇)‘𝐴, the eigenvalue associated with eigenvector 𝐴, is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
⊢ eigval = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2))))
Definition	df-spec 32188*	Define the spectrum of an operator. Definition of spectrum in [Halmos] p. 50. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
⊢ Lambda = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
20.6.10  Theorems about operators and functionals
Theorem	nmopval 32189*	Value of the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
Theorem	elcnop 32190*	Property defining a continuous Hilbert space operator. (Contributed by NM, 28-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ (𝑇 ∈ ContOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑇‘𝑤) −ℎ (𝑇‘𝑥))) < 𝑦)))
Theorem	ellnop 32191*	Property defining a linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ (𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧))))
Theorem	lnopf 32192	A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
Theorem	elbdop 32193	Property defining a bounded linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞))
Theorem	bdopln 32194	A bounded linear Hilbert space operator is a linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)
Theorem	bdopf 32195	A bounded linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
Theorem	nmopsetretALT 32196*	The set in the supremum of the operator norm definition df-nmop 32172 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ)
Theorem	nmopsetretHIL 32197*	The set in the supremum of the operator norm definition df-nmop 32172 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ)
Theorem	nmopsetn0 32198*	The set in the supremum of the operator norm definition df-nmop 32172 is nonempty. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
⊢ (normℎ‘(𝑇‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}
Theorem	nmopxr 32199	The norm of a Hilbert space operator is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) ∈ ℝ*)
Theorem	nmoprepnf 32200	The norm of a Hilbert space operator is either real or plus infinity. (Contributed by NM, 5-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → ((normop‘𝑇) ∈ ℝ ↔ (normop‘𝑇) ≠ +∞))