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Theorem List for Metamath Proof Explorer - 28101-28200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnlnadjlem4 28101* Lemma for cnlnadji 28107. The values of auxiliary function 𝐹 are vectors. (Contributed by NM, 17-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑇 ∈ ConOp    &   𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))    &   𝐵 = (𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)       (𝐴 ∈ ℋ → (𝐹𝐴) ∈ ℋ)
 
Theoremcnlnadjlem5 28102* Lemma for cnlnadji 28107. 𝐹 is an adjoint of 𝑇 (later, we will show it is unique). (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑇 ∈ ConOp    &   𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))    &   𝐵 = (𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)       ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇𝐶) ·ih 𝐴) = (𝐶 ·ih (𝐹𝐴)))
 
Theoremcnlnadjlem6 28103* Lemma for cnlnadji 28107. 𝐹 is linear. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑇 ∈ ConOp    &   𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))    &   𝐵 = (𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)       𝐹 ∈ LinOp
 
Theoremcnlnadjlem7 28104* Lemma for cnlnadji 28107. Helper lemma to show that 𝐹 is continuous. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑇 ∈ ConOp    &   𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))    &   𝐵 = (𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)       (𝐴 ∈ ℋ → (norm‘(𝐹𝐴)) ≤ ((normop𝑇) · (norm𝐴)))
 
Theoremcnlnadjlem8 28105* Lemma for cnlnadji 28107. 𝐹 is continuous. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑇 ∈ ConOp    &   𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))    &   𝐵 = (𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)       𝐹 ∈ ConOp
 
Theoremcnlnadjlem9 28106* Lemma for cnlnadji 28107. 𝐹 provides an example showing the existence of a continuous linear adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑇 ∈ ConOp    &   𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))    &   𝐵 = (𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)       𝑡 ∈ (LinOp ∩ ConOp)∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑇𝑥) ·ih 𝑧) = (𝑥 ·ih (𝑡𝑧))
 
Theoremcnlnadji 28107* Every continuous linear operator has an adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑇 ∈ ConOp       𝑡 ∈ (LinOp ∩ ConOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡𝑦))
 
Theoremcnlnadjeui 28108* Every continuous linear operator has a unique adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑇 ∈ ConOp       ∃!𝑡 ∈ (LinOp ∩ ConOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡𝑦))
 
Theoremcnlnadjeu 28109* Every continuous linear operator has a unique adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ (LinOp ∩ ConOp) → ∃!𝑡 ∈ (LinOp ∩ ConOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡𝑦)))
 
Theoremcnlnadj 28110* Every continuous linear operator has an adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ (LinOp ∩ ConOp) → ∃𝑡 ∈ (LinOp ∩ ConOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡𝑦)))
 
Theoremcnlnssadj 28111 Every continuous linear Hilbert space operator has an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
(LinOp ∩ ConOp) ⊆ dom adj
 
Theorembdopssadj 28112 Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
BndLinOp ⊆ dom adj
 
Theorembdopadj 28113 Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp → 𝑇 ∈ dom adj)
 
Theoremadjbdln 28114 The adjoint of a bounded linear operator is a bounded linear operator. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp → (adj𝑇) ∈ BndLinOp)
 
Theoremadjbdlnb 28115 An operator is bounded and linear iff its adjoint is. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp ↔ (adj𝑇) ∈ BndLinOp)
 
Theoremadjbd1o 28116 The mapping of adjoints of bounded linear operators is one-to-one onto. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
(adj ↾ BndLinOp):BndLinOp–1-1-onto→BndLinOp
 
Theoremadjlnop 28117 The adjoint of an operator is linear. Proposition 1 of [AkhiezerGlazman] p. 80. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)
(𝑇 ∈ dom adj → (adj𝑇) ∈ LinOp)
 
Theoremadjsslnop 28118 Every operator with an adjoint is linear. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)
dom adj ⊆ LinOp
 
Theoremnmopadjlei 28119 Property of the norm of an adjoint. Part of proof of Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ BndLinOp       (𝐴 ∈ ℋ → (norm‘((adj𝑇)‘𝐴)) ≤ ((normop𝑇) · (norm𝐴)))
 
Theoremnmopadjlem 28120 Lemma for nmopadji 28121. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ BndLinOp       (normop‘(adj𝑇)) ≤ (normop𝑇)
 
Theoremnmopadji 28121 Property of the norm of an adjoint. Theorem 3.11(v) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ BndLinOp       (normop‘(adj𝑇)) = (normop𝑇)
 
Theoremadjeq0 28122 An operator is zero iff its adjoint is zero. Theorem 3.11(i) of [Beran] p. 106. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
(𝑇 = 0hop ↔ (adj𝑇) = 0hop )
 
Theoremadjmul 28123 The adjoint of the scalar product of an operator. Theorem 3.11(ii) of [Beran] p. 106. (Contributed by NM, 21-Feb-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adj) → (adj‘(𝐴 ·op 𝑇)) = ((∗‘𝐴) ·op (adj𝑇)))
 
Theoremadjadd 28124 The adjoint of the sum of two operators. Theorem 3.11(iii) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (adj‘(𝑆 +op 𝑇)) = ((adj𝑆) +op (adj𝑇)))
 
Theoremnmoptrii 28125 Triangle inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ BndLinOp    &   𝑇 ∈ BndLinOp       (normop‘(𝑆 +op 𝑇)) ≤ ((normop𝑆) + (normop𝑇))
 
Theoremnmopcoi 28126 Upper bound for the norm of the composition of two bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ BndLinOp    &   𝑇 ∈ BndLinOp       (normop‘(𝑆𝑇)) ≤ ((normop𝑆) · (normop𝑇))
 
Theorembdophsi 28127 The sum of two bounded linear operators is a bounded linear operator. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ BndLinOp    &   𝑇 ∈ BndLinOp       (𝑆 +op 𝑇) ∈ BndLinOp
 
Theorembdophdi 28128 The difference between two bounded linear operators is bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ BndLinOp    &   𝑇 ∈ BndLinOp       (𝑆op 𝑇) ∈ BndLinOp
 
Theorembdopcoi 28129 The composition of two bounded linear operators is bounded. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ BndLinOp    &   𝑇 ∈ BndLinOp       (𝑆𝑇) ∈ BndLinOp
 
Theoremnmoptri2i 28130 Triangle-type inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ BndLinOp    &   𝑇 ∈ BndLinOp       ((normop𝑆) − (normop𝑇)) ≤ (normop‘(𝑆 +op 𝑇))
 
Theoremadjcoi 28131 The adjoint of a composition of bounded linear operators. Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ BndLinOp    &   𝑇 ∈ BndLinOp       (adj‘(𝑆𝑇)) = ((adj𝑇) ∘ (adj𝑆))
 
Theoremnmopcoadji 28132 The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)
𝑇 ∈ BndLinOp       (normop‘((adj𝑇) ∘ 𝑇)) = ((normop𝑇)↑2)
 
Theoremnmopcoadj2i 28133 The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑇 ∈ BndLinOp       (normop‘(𝑇 ∘ (adj𝑇))) = ((normop𝑇)↑2)
 
Theoremnmopcoadj0i 28134 An operator composed with its adjoint is zero iff the operator is zero. Theorem 3.11(vii) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑇 ∈ BndLinOp       ((𝑇 ∘ (adj𝑇)) = 0hop𝑇 = 0hop )
 
19.6.13  Quantum computation error bound theorem
 
Theoremunierri 28135 If we approximate a chain of unitary transformations (quantum computer gates) 𝐹, 𝐺 by other unitary transformations 𝑆, 𝑇, the error increases at most additively. Equation 4.73 of [NielsenChuang] p. 195. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝐹 ∈ UniOp    &   𝐺 ∈ UniOp    &   𝑆 ∈ UniOp    &   𝑇 ∈ UniOp       (normop‘((𝐹𝐺) −op (𝑆𝑇))) ≤ ((normop‘(𝐹op 𝑆)) + (normop‘(𝐺op 𝑇)))
 
19.6.14  Dirac bra-ket notation (cont.)
 
Theorembranmfn 28136 The norm of the bra function. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (norm𝐴))
 
Theorembrabn 28137 The bra of a vector is a bounded functional. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) ∈ ℝ)
 
Theoremrnbra 28138 The set of bras equals the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
ran bra = (LinFn ∩ ConFn)
 
Theorembra11 28139 The bra function maps vectors one-to-one onto the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
bra: ℋ–1-1-onto→(LinFn ∩ ConFn)
 
Theorembracnln 28140 A bra is a continuous linear functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (bra‘𝐴) ∈ (LinFn ∩ ConFn))
 
Theoremcnvbraval 28141* Value of the converse of the bra function. Based on the Riesz Lemma riesz4 28095, this very important theorem not only justifies the Dirac bra-ket notation, but allows us to extract a unique vector from any continuous linear functional from which the functional can be recovered; i.e. a single vector can "store" all of the information contained in any entire continuous linear functional (mapping from to ). (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
(𝑇 ∈ (LinFn ∩ ConFn) → (bra‘𝑇) = (𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑥 ·ih 𝑦)))
 
Theoremcnvbracl 28142 Closure of the converse of the bra function. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
(𝑇 ∈ (LinFn ∩ ConFn) → (bra‘𝑇) ∈ ℋ)
 
Theoremcnvbrabra 28143 The converse bra of the bra of a vector is the vector itself. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (bra‘(bra‘𝐴)) = 𝐴)
 
Theorembracnvbra 28144 The bra of the converse bra of a continuous linear functional. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
(𝑇 ∈ (LinFn ∩ ConFn) → (bra‘(bra‘𝑇)) = 𝑇)
 
Theorembracnlnval 28145* The vector that a continuous linear functional is the bra of. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
(𝑇 ∈ (LinFn ∩ ConFn) → 𝑇 = (bra‘(𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑥 ·ih 𝑦))))
 
Theoremcnvbramul 28146 Multiplication property of the converse bra function. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇 ∈ (LinFn ∩ ConFn)) → (bra‘(𝐴 ·fn 𝑇)) = ((∗‘𝐴) · (bra‘𝑇)))
 
Theoremkbass1 28147 Dirac bra-ket associative law ( ∣ 𝐴 𝐵 ∣ ) ∣ 𝐶⟩ = 𝐴⟩(⟨𝐵𝐶⟩) i.e. the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since 𝐵𝐶 is a complex number, it is the first argument in the inner product · that it is mapped to, although in Dirac notation it is placed after the ket. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = (((bra‘𝐵)‘𝐶) · 𝐴))
 
Theoremkbass2 28148 Dirac bra-ket associative law (⟨𝐴𝐵⟩)⟨𝐶 ∣ = 𝐴 ∣ ( ∣ 𝐵 𝐶 ∣ ) i.e. the juxtaposition of an inner product with a bra equals a ket juxtaposed with an outer product. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((bra‘𝐴)‘𝐵) ·fn (bra‘𝐶)) = ((bra‘𝐴) ∘ (𝐵 ketbra 𝐶)))
 
Theoremkbass3 28149 Dirac bra-ket associative law 𝐴𝐵 𝐶𝐷⟩ = (⟨𝐴𝐵 𝐶 ∣ ) ∣ 𝐷. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((bra‘𝐴)‘𝐵) · ((bra‘𝐶)‘𝐷)) = ((((bra‘𝐴)‘𝐵) ·fn (bra‘𝐶))‘𝐷))
 
Theoremkbass4 28150 Dirac bra-ket associative law 𝐴𝐵 𝐶𝐷⟩ = 𝐴 ∣ ( ∣ 𝐵 𝐶𝐷⟩). (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((bra‘𝐴)‘𝐵) · ((bra‘𝐶)‘𝐷)) = ((bra‘𝐴)‘(((bra‘𝐶)‘𝐷) · 𝐵)))
 
Theoremkbass5 28151 Dirac bra-ket associative law ( ∣ 𝐴 𝐵 ∣ )( ∣ 𝐶 𝐷 ∣ ) = (( ∣ 𝐴 𝐵 ∣ ) ∣ 𝐶⟩)⟨𝐷. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷))
 
Theoremkbass6 28152 Dirac bra-ket associative law ( ∣ 𝐴 𝐵 ∣ )( ∣ 𝐶 𝐷 ∣ ) = ∣ 𝐴 (⟨𝐵 ∣ ( ∣ 𝐶 𝐷 ∣ )). (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (𝐴 ketbra (bra‘((bra‘𝐵) ∘ (𝐶 ketbra 𝐷)))))
 
19.6.15  Positive operators (cont.)
 
Theoremleopg 28153* Ordering relation for positive operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝑇𝐴𝑈𝐵) → (𝑇op 𝑈 ↔ ((𝑈op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥))))
 
Theoremleop 28154* Ordering relation for operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇op 𝑈 ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥)))
 
Theoremleop2 28155* Ordering relation for operators. Definition of operator ordering in [Young] p. 141. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑇𝑥) ·ih 𝑥) ≤ ((𝑈𝑥) ·ih 𝑥)))
 
Theoremleop3 28156 Operator ordering in terms of a positive operator. Definition of operator ordering in [Retherford] p. 49. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇op 𝑈 ↔ 0hopop (𝑈op 𝑇)))
 
Theoremleoppos 28157* Binary relation defining a positive operator. Definition VI.1 of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → ( 0hopop 𝑇 ↔ ∀𝑥 ∈ ℋ 0 ≤ ((𝑇𝑥) ·ih 𝑥)))
 
Theoremleoprf2 28158 The ordering relation for operators is reflexive. (Contributed by NM, 24-Jul-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → 𝑇op 𝑇)
 
Theoremleoprf 28159 The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇op 𝑇)
 
Theoremleopsq 28160 The square of a Hermitian operator is positive. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 0hopop (𝑇𝑇))
 
Theorem0leop 28161 The zero operator is a positive operator. (The literature calls it "positive," even though in some sense it is really "nonnegative.") Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
0hopop 0hop
 
Theoremidleop 28162 The identity operator is a positive operator. Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
0hopop Iop
 
Theoremleopadd 28163 The sum of two positive operators is positive. Exercise 1(i) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
(((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ ( 0hopop 𝑇 ∧ 0hopop 𝑈)) → 0hopop (𝑇 +op 𝑈))
 
Theoremleopmuli 28164 The scalar product of a nonnegative real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
(((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 0hopop 𝑇)) → 0hopop (𝐴 ·op 𝑇))
 
Theoremleopmul 28165 The scalar product of a positive real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 0 < 𝐴) → ( 0hopop 𝑇 ↔ 0hopop (𝐴 ·op 𝑇)))
 
Theoremleopmul2i 28166 Scalar product applied to operator ordering. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
(((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (0 ≤ 𝐴𝑇op 𝑈)) → (𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈))
 
Theoremleoptri 28167 The positive operator ordering relation satisfies trichotomy. Exercise 1(iii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ((𝑇op 𝑈𝑈op 𝑇) ↔ 𝑇 = 𝑈))
 
Theoremleoptr 28168 The positive operator ordering relation is transitive. Exercise 1(iv) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
(((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑆op 𝑇𝑇op 𝑈)) → 𝑆op 𝑈)
 
Theoremleopnmid 28169 A bounded Hermitian operator is less than or equal to its norm times the identity operator. (Contributed by NM, 11-Aug-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ (normop𝑇) ∈ ℝ) → 𝑇op ((normop𝑇) ·op Iop ))
 
Theoremnmopleid 28170 A nonzero, bounded Hermitian operator divided by its norm is less than or equal to the identity operator. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ (normop𝑇) ∈ ℝ ∧ 𝑇 ≠ 0hop ) → ((1 / (normop𝑇)) ·op 𝑇) ≤op Iop )
 
Theoremopsqrlem1 28171* Lemma for opsqri . (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   (normop𝑇) ∈ ℝ    &    0hopop 𝑇    &   𝑅 = ((1 / (normop𝑇)) ·op 𝑇)    &   (𝑇 ≠ 0hop → ∃𝑢 ∈ HrmOp ( 0hopop 𝑢 ∧ (𝑢𝑢) = 𝑅))       (𝑇 ≠ 0hop → ∃𝑣 ∈ HrmOp ( 0hopop 𝑣 ∧ (𝑣𝑣) = 𝑇))
 
Theoremopsqrlem2 28172* Lemma for opsqri . 𝐹𝑁 is the recursive function An (starting at n=1 instead of 0) of Theorem 9.4-2 of [Kreyszig] p. 476. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))       (𝐹‘1) = 0hop
 
Theoremopsqrlem3 28173* Lemma for opsqri . (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))       ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))
 
Theoremopsqrlem4 28174* Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))       𝐹:ℕ⟶HrmOp
 
Theoremopsqrlem5 28175* Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))       (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) = ((𝐹𝑁) +op ((1 / 2) ·op (𝑇op ((𝐹𝑁) ∘ (𝐹𝑁))))))
 
Theoremopsqrlem6 28176* Lemma for opsqri . (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))    &   𝑇op Iop       (𝑁 ∈ ℕ → (𝐹𝑁) ≤op Iop )
 
19.6.16  Projectors as operators
 
Theorempjhmopi 28177 A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻) ∈ HrmOp
 
Theorempjlnopi 28178 A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻) ∈ LinOp
 
Theorempjnmopi 28179 The operator norm of a projector on a nonzero closed subspace is one. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
𝐻C       (𝐻 ≠ 0 → (normop‘(proj𝐻)) = 1)
 
Theorempjbdlni 28180 A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻) ∈ BndLinOp
 
Theorempjhmop 28181 A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → (proj𝐻) ∈ HrmOp)
 
Theoremhmopidmchi 28182 An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 21-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   (𝑇𝑇) = 𝑇       ran 𝑇C
 
Theoremhmopidmpji 28183 An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Halmos seems to omit the proof that 𝐻 is a closed subspace, which is not trivial as hmopidmchi 28182 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   (𝑇𝑇) = 𝑇       𝑇 = (proj‘ran 𝑇)
 
Theoremhmopidmch 28184 An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ (𝑇𝑇) = 𝑇) → ran 𝑇C )
 
Theoremhmopidmpj 28185 An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ (𝑇𝑇) = 𝑇) → 𝑇 = (proj‘ran 𝑇))
 
Theorempjsdii 28186 Distributive law for Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
𝐻C    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((proj𝐻) ∘ (𝑆 +op 𝑇)) = (((proj𝐻) ∘ 𝑆) +op ((proj𝐻) ∘ 𝑇))
 
Theorempjddii 28187 Distributive law for Hilbert space operator difference. (Contributed by NM, 24-Nov-2000.) (New usage is discouraged.)
𝐻C    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((proj𝐻) ∘ (𝑆op 𝑇)) = (((proj𝐻) ∘ 𝑆) −op ((proj𝐻) ∘ 𝑇))
 
Theorempjsdi2i 28188 Chained distributive law for Hilbert space operator difference. (Contributed by NM, 30-Nov-2000.) (New usage is discouraged.)
𝐻C    &   𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅 ∘ (𝑆 +op 𝑇)) = ((𝑅𝑆) +op (𝑅𝑇)) → (((proj𝐻) ∘ 𝑅) ∘ (𝑆 +op 𝑇)) = ((((proj𝐻) ∘ 𝑅) ∘ 𝑆) +op (((proj𝐻) ∘ 𝑅) ∘ 𝑇)))
 
Theorempjcoi 28189 Composition of projections. (Contributed by NM, 16-Aug-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (((proj𝐺) ∘ (proj𝐻))‘𝐴) = ((proj𝐺)‘((proj𝐻)‘𝐴)))
 
Theorempjcocli 28190 Closure of composition of projections. (Contributed by NM, 29-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (((proj𝐺) ∘ (proj𝐻))‘𝐴) ∈ 𝐺)
 
Theorempjcohcli 28191 Closure of composition of projections. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (((proj𝐺) ∘ (proj𝐻))‘𝐴) ∈ ℋ)
 
Theorempjadjcoi 28192 Adjoint of composition of projections. Special case of Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 6-Oct-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((((proj𝐺) ∘ (proj𝐻))‘𝐴) ·ih 𝐵) = (𝐴 ·ih (((proj𝐻) ∘ (proj𝐺))‘𝐵)))
 
Theorempjcofni 28193 Functionality of composition of projections. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       ((proj𝐺) ∘ (proj𝐻)) Fn ℋ
 
Theorempjss1coi 28194 Subset relationship for projections. Theorem 4.5(i)<->(iii) of [Beran] p. 112. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ((proj𝐻) ∘ (proj𝐺)) = (proj𝐺))
 
Theorempjss2coi 28195 Subset relationship for projections. Theorem 4.5(i)<->(ii) of [Beran] p. 112. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ((proj𝐺) ∘ (proj𝐻)) = (proj𝐺))
 
Theorempjssmi 28196 Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (𝐻𝐺 → (((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) = ((proj‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴)))
 
Theorempjssge0i 28197 Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) = ((proj‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) → 0 ≤ ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) ·ih 𝐴)))
 
Theorempjdifnormi 28198 Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (0 ≤ ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) ·ih 𝐴) ↔ (norm‘((proj𝐻)‘𝐴)) ≤ (norm‘((proj𝐺)‘𝐴))))
 
Theorempjnormssi 28199* Theorem 4.5(i)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ∀𝑥 ∈ ℋ (norm‘((proj𝐺)‘𝑥)) ≤ (norm‘((proj𝐻)‘𝑥)))
 
Theorempjorthcoi 28200 Composition of projections of orthogonal subspaces. Part (i)->(iia) of Theorem 27.4 of [Halmos] p. 45. (Contributed by NM, 6-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺 ⊆ (⊥‘𝐻) → ((proj𝐺) ∘ (proj𝐻)) = 0hop )
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