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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cnlnadjlem2 32001* | Lemma for cnlnadji 32009. 𝐺 is a continuous linear functional. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.) |
⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) ⇒ ⊢ (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn)) | ||
Theorem | cnlnadjlem3 32002* | Lemma for cnlnadji 32009. By riesz4 31997, 𝐵 is the unique vector such that (𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤) for all 𝑣. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.) |
⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) ⇒ ⊢ (𝑦 ∈ ℋ → 𝐵 ∈ ℋ) | ||
Theorem | cnlnadjlem4 32003* | Lemma for cnlnadji 32009. 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 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ (𝐴 ∈ ℋ → (𝐹‘𝐴) ∈ ℋ) | ||
Theorem | cnlnadjlem5 32004* | Lemma for cnlnadji 32009. 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 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐶) ·ih 𝐴) = (𝐶 ·ih (𝐹‘𝐴))) | ||
Theorem | cnlnadjlem6 32005* | Lemma for cnlnadji 32009. 𝐹 is linear. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.) |
⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ 𝐹 ∈ LinOp | ||
Theorem | cnlnadjlem7 32006* | Lemma for cnlnadji 32009. Helper lemma to show that 𝐹 is continuous. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝐹‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
Theorem | cnlnadjlem8 32007* | Lemma for cnlnadji 32009. 𝐹 is continuous. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.) |
⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ 𝐹 ∈ ContOp | ||
Theorem | cnlnadjlem9 32008* | Lemma for cnlnadji 32009. 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 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑧) = (𝑥 ·ih (𝑡‘𝑧)) | ||
Theorem | cnlnadji 32009* | 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 ∈ ContOp ⇒ ⊢ ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦)) | ||
Theorem | cnlnadjeui 32010* | 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 ∈ ContOp ⇒ ⊢ ∃!𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦)) | ||
Theorem | cnlnadjeu 32011* | 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 ∩ ContOp) → ∃!𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦))) | ||
Theorem | cnlnadj 32012* | 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 ∩ ContOp) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦))) | ||
Theorem | cnlnssadj 32013 | Every continuous linear Hilbert space operator has an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
⊢ (LinOp ∩ ContOp) ⊆ dom adjℎ | ||
Theorem | bdopssadj 32014 | Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
⊢ BndLinOp ⊆ dom adjℎ | ||
Theorem | bdopadj 32015 | Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ dom adjℎ) | ||
Theorem | adjbdln 32016 | The adjoint of a bounded linear operator is a bounded linear operator. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ BndLinOp → (adjℎ‘𝑇) ∈ BndLinOp) | ||
Theorem | adjbdlnb 32017 | An operator is bounded and linear iff its adjoint is. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ BndLinOp ↔ (adjℎ‘𝑇) ∈ BndLinOp) | ||
Theorem | adjbd1o 32018 | The mapping of adjoints of bounded linear operators is one-to-one onto. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
⊢ (adjℎ ↾ BndLinOp):BndLinOp–1-1-onto→BndLinOp | ||
Theorem | adjlnop 32019 | 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 adjℎ → (adjℎ‘𝑇) ∈ LinOp) | ||
Theorem | adjsslnop 32020 | Every operator with an adjoint is linear. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.) |
⊢ dom adjℎ ⊆ LinOp | ||
Theorem | nmopadjlei 32021 | 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 → (normℎ‘((adjℎ‘𝑇)‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
Theorem | nmopadjlem 32022 | Lemma for nmopadji 32023. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇) | ||
Theorem | nmopadji 32023 | 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‘(adjℎ‘𝑇)) = (normop‘𝑇) | ||
Theorem | adjeq0 32024 | 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 ↔ (adjℎ‘𝑇) = 0hop ) | ||
Theorem | adjmul 32025 | 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 adjℎ) → (adjℎ‘(𝐴 ·op 𝑇)) = ((∗‘𝐴) ·op (adjℎ‘𝑇))) | ||
Theorem | adjadd 32026 | 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 adjℎ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝑆 +op 𝑇)) = ((adjℎ‘𝑆) +op (adjℎ‘𝑇))) | ||
Theorem | nmoptrii 32027 | 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)) ≤ ((normop‘𝑆) + (normop‘𝑇)) | ||
Theorem | nmopcoi 32028 | 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‘(𝑆 ∘ 𝑇)) ≤ ((normop‘𝑆) · (normop‘𝑇)) | ||
Theorem | bdophsi 32029 | 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 | ||
Theorem | bdophdi 32030 | The difference between two bounded linear operators is bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝑆 −op 𝑇) ∈ BndLinOp | ||
Theorem | bdopcoi 32031 | The composition of two bounded linear operators is bounded. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝑆 ∘ 𝑇) ∈ BndLinOp | ||
Theorem | nmoptri2i 32032 | Triangle-type inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ ((normop‘𝑆) − (normop‘𝑇)) ≤ (normop‘(𝑆 +op 𝑇)) | ||
Theorem | adjcoi 32033 | 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 ⇒ ⊢ (adjℎ‘(𝑆 ∘ 𝑇)) = ((adjℎ‘𝑇) ∘ (adjℎ‘𝑆)) | ||
Theorem | nmopcoadji 32034 | 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‘((adjℎ‘𝑇) ∘ 𝑇)) = ((normop‘𝑇)↑2) | ||
Theorem | nmopcoadj2i 32035 | 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 ∘ (adjℎ‘𝑇))) = ((normop‘𝑇)↑2) | ||
Theorem | nmopcoadj0i 32036 | 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 ∘ (adjℎ‘𝑇)) = 0hop ↔ 𝑇 = 0hop ) | ||
Theorem | unierri 32037 | 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‘((𝐹 ∘ 𝐺) −op (𝑆 ∘ 𝑇))) ≤ ((normop‘(𝐹 −op 𝑆)) + (normop‘(𝐺 −op 𝑇))) | ||
Theorem | branmfn 32038 | The norm of the bra function. (Contributed by NM, 24-May-2006.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (normℎ‘𝐴)) | ||
Theorem | brabn 32039 | 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) | ||
Theorem | rnbra 32040 | The set of bras equals the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
⊢ ran bra = (LinFn ∩ ContFn) | ||
Theorem | bra11 32041 | 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 ∩ ContFn) | ||
Theorem | bracnln 32042 | A bra is a continuous linear functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (bra‘𝐴) ∈ (LinFn ∩ ContFn)) | ||
Theorem | cnvbraval 32043* | Value of the converse of the bra function. Based on the Riesz Lemma riesz4 31997, this very important theorem not only justifies the Dirac bra-ket notation, but allows 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 ℋ to ℂ). (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ (LinFn ∩ ContFn) → (◡bra‘𝑇) = (℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦))) | ||
Theorem | cnvbracl 32044 | Closure of the converse of the bra function. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ (LinFn ∩ ContFn) → (◡bra‘𝑇) ∈ ℋ) | ||
Theorem | cnvbrabra 32045 | 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) | ||
Theorem | bracnvbra 32046 | The bra of the converse bra of a continuous linear functional. (Contributed by NM, 31-May-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ (LinFn ∩ ContFn) → (bra‘(◡bra‘𝑇)) = 𝑇) | ||
Theorem | bracnlnval 32047* | The vector that a continuous linear functional is the bra of. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ (LinFn ∩ ContFn) → 𝑇 = (bra‘(℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦)))) | ||
Theorem | cnvbramul 32048 | Multiplication property of the converse bra function. (Contributed by NM, 31-May-2006.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ (LinFn ∩ ContFn)) → (◡bra‘(𝐴 ·fn 𝑇)) = ((∗‘𝐴) ·ℎ (◡bra‘𝑇))) | ||
Theorem | kbass1 32049 | Dirac bra-ket associative law ( ∣ 𝐴〉〈𝐵 ∣ ) ∣ 𝐶〉 = ∣ 𝐴〉(〈𝐵 ∣ 𝐶〉), i.e., the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since 〈𝐵 ∣ 𝐶〉 is a complex number, it is the first argument in the inner product ·ℎ 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) ·ℎ 𝐴)) | ||
Theorem | kbass2 32050 | Dirac bra-ket associative law (〈𝐴 ∣ 𝐵〉)〈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))) | ||
Theorem | kbass3 32051 | 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)) → (((bra‘𝐴)‘𝐵) · ((bra‘𝐶)‘𝐷)) = ((((bra‘𝐴)‘𝐵) ·fn (bra‘𝐶))‘𝐷)) | ||
Theorem | kbass4 32052 | 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)) → (((bra‘𝐴)‘𝐵) · ((bra‘𝐶)‘𝐷)) = ((bra‘𝐴)‘(((bra‘𝐶)‘𝐷) ·ℎ 𝐵))) | ||
Theorem | kbass5 32053 | 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)) | ||
Theorem | kbass6 32054 | 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))))) | ||
Theorem | leopg 32055* | Ordering relation for positive operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
⊢ ((𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐵) → (𝑇 ≤op 𝑈 ↔ ((𝑈 −op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥)))) | ||
Theorem | leop 32056* | 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) → (𝑇 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥))) | ||
Theorem | leop2 32057* | 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) → (𝑇 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) | ||
Theorem | leop3 32058 | 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) → (𝑇 ≤op 𝑈 ↔ 0hop ≤op (𝑈 −op 𝑇))) | ||
Theorem | leoppos 32059* | 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 → ( 0hop ≤op 𝑇 ↔ ∀𝑥 ∈ ℋ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥))) | ||
Theorem | leoprf2 32060 | The ordering relation for operators is reflexive. (Contributed by NM, 24-Jul-2006.) (New usage is discouraged.) |
⊢ (𝑇: ℋ⟶ ℋ → 𝑇 ≤op 𝑇) | ||
Theorem | leoprf 32061 | The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ HrmOp → 𝑇 ≤op 𝑇) | ||
Theorem | leopsq 32062 | The square of a Hermitian operator is positive. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ HrmOp → 0hop ≤op (𝑇 ∘ 𝑇)) | ||
Theorem | 0leop 32063 | 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.) |
⊢ 0hop ≤op 0hop | ||
Theorem | idleop 32064 | 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.) |
⊢ 0hop ≤op Iop | ||
Theorem | leopadd 32065 | 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) ∧ ( 0hop ≤op 𝑇 ∧ 0hop ≤op 𝑈)) → 0hop ≤op (𝑇 +op 𝑈)) | ||
Theorem | leopmuli 32066 | 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 ∧ 0hop ≤op 𝑇)) → 0hop ≤op (𝐴 ·op 𝑇)) | ||
Theorem | leopmul 32067 | 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) → ( 0hop ≤op 𝑇 ↔ 0hop ≤op (𝐴 ·op 𝑇))) | ||
Theorem | leopmul2i 32068 | Scalar product applied to operator ordering. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 𝑇 ≤op 𝑈)) → (𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈)) | ||
Theorem | leoptri 32069 | 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) → ((𝑇 ≤op 𝑈 ∧ 𝑈 ≤op 𝑇) ↔ 𝑇 = 𝑈)) | ||
Theorem | leoptr 32070 | 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) ∧ (𝑆 ≤op 𝑇 ∧ 𝑇 ≤op 𝑈)) → 𝑆 ≤op 𝑈) | ||
Theorem | leopnmid 32071 | 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 ∧ (normop‘𝑇) ∈ ℝ) → 𝑇 ≤op ((normop‘𝑇) ·op Iop )) | ||
Theorem | nmopleid 32072 | 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 ∧ (normop‘𝑇) ∈ ℝ ∧ 𝑇 ≠ 0hop ) → ((1 / (normop‘𝑇)) ·op 𝑇) ≤op Iop ) | ||
Theorem | opsqrlem1 32073* | Lemma for opsqri . (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.) |
⊢ 𝑇 ∈ HrmOp & ⊢ (normop‘𝑇) ∈ ℝ & ⊢ 0hop ≤op 𝑇 & ⊢ 𝑅 = ((1 / (normop‘𝑇)) ·op 𝑇) & ⊢ (𝑇 ≠ 0hop → ∃𝑢 ∈ HrmOp ( 0hop ≤op 𝑢 ∧ (𝑢 ∘ 𝑢) = 𝑅)) ⇒ ⊢ (𝑇 ≠ 0hop → ∃𝑣 ∈ HrmOp ( 0hop ≤op 𝑣 ∧ (𝑣 ∘ 𝑣) = 𝑇)) | ||
Theorem | opsqrlem2 32074* | Lemma for opsqri . 𝐹‘𝑁 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 (𝑇 −op (𝑥 ∘ 𝑥))))) & ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) ⇒ ⊢ (𝐹‘1) = 0hop | ||
Theorem | opsqrlem3 32075* | 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 (𝑇 −op (𝑥 ∘ 𝑥))))) & ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) ⇒ ⊢ ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺))))) | ||
Theorem | opsqrlem4 32076* | 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 (𝑇 −op (𝑥 ∘ 𝑥))))) & ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) ⇒ ⊢ 𝐹:ℕ⟶HrmOp | ||
Theorem | opsqrlem5 32077* | 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 (𝑇 −op (𝑥 ∘ 𝑥))))) & ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) ⇒ ⊢ (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) = ((𝐹‘𝑁) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑁) ∘ (𝐹‘𝑁)))))) | ||
Theorem | opsqrlem6 32078* | 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 (𝑇 −op (𝑥 ∘ 𝑥))))) & ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) & ⊢ 𝑇 ≤op Iop ⇒ ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤op Iop ) | ||
Theorem | pjhmopi 32079 | A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻) ∈ HrmOp | ||
Theorem | pjlnopi 32080 | A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻) ∈ LinOp | ||
Theorem | pjnmopi 32081 | 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.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐻 ≠ 0ℋ → (normop‘(projℎ‘𝐻)) = 1) | ||
Theorem | pjbdlni 32082 | A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻) ∈ BndLinOp | ||
Theorem | pjhmop 32083 | A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) ∈ HrmOp) | ||
Theorem | hmopidmchi 32084 | 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 & ⊢ (𝑇 ∘ 𝑇) = 𝑇 ⇒ ⊢ ran 𝑇 ∈ Cℋ | ||
Theorem | hmopidmpji 32085 | 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 32084 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
⊢ 𝑇 ∈ HrmOp & ⊢ (𝑇 ∘ 𝑇) = 𝑇 ⇒ ⊢ 𝑇 = (projℎ‘ran 𝑇) | ||
Theorem | hmopidmch 32086 | 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 ∧ (𝑇 ∘ 𝑇) = 𝑇) → ran 𝑇 ∈ Cℋ ) | ||
Theorem | hmopidmpj 32087 | 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 ∧ (𝑇 ∘ 𝑇) = 𝑇) → 𝑇 = (projℎ‘ran 𝑇)) | ||
Theorem | pjsdii 32088 | Distributive law for Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((projℎ‘𝐻) ∘ (𝑆 +op 𝑇)) = (((projℎ‘𝐻) ∘ 𝑆) +op ((projℎ‘𝐻) ∘ 𝑇)) | ||
Theorem | pjddii 32089 | Distributive law for Hilbert space operator difference. (Contributed by NM, 24-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((projℎ‘𝐻) ∘ (𝑆 −op 𝑇)) = (((projℎ‘𝐻) ∘ 𝑆) −op ((projℎ‘𝐻) ∘ 𝑇)) | ||
Theorem | pjsdi2i 32090 | Chained distributive law for Hilbert space operator difference. (Contributed by NM, 30-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ & ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((𝑅 ∘ (𝑆 +op 𝑇)) = ((𝑅 ∘ 𝑆) +op (𝑅 ∘ 𝑇)) → (((projℎ‘𝐻) ∘ 𝑅) ∘ (𝑆 +op 𝑇)) = ((((projℎ‘𝐻) ∘ 𝑅) ∘ 𝑆) +op (((projℎ‘𝐻) ∘ 𝑅) ∘ 𝑇))) | ||
Theorem | pjcoi 32091 | Composition of projections. (Contributed by NM, 16-Aug-2000.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) = ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝐴))) | ||
Theorem | pjcocli 32092 | Closure of composition of projections. (Contributed by NM, 29-Nov-2000.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) ∈ 𝐺) | ||
Theorem | pjcohcli 32093 | Closure of composition of projections. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) ∈ ℋ) | ||
Theorem | pjadjcoi 32094 | 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.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝐴) ·ih 𝐵) = (𝐴 ·ih (((projℎ‘𝐻) ∘ (projℎ‘𝐺))‘𝐵))) | ||
Theorem | pjcofni 32095 | Functionality of composition of projections. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) Fn ℋ | ||
Theorem | pjss1coi 32096 | Subset relationship for projections. Theorem 4.5(i)<->(iii) of [Beran] p. 112. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) = (projℎ‘𝐺)) | ||
Theorem | pjss2coi 32097 | Subset relationship for projections. Theorem 4.5(i)<->(ii) of [Beran] p. 112. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘𝐺)) | ||
Theorem | pjssmi 32098 | 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.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → (𝐻 ⊆ 𝐺 → (((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) = ((projℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴))) | ||
Theorem | pjssge0i 32099 | Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → ((((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) = ((projℎ‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) → 0 ≤ ((((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) ·ih 𝐴))) | ||
Theorem | pjdifnormi 32100 | Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.) |
⊢ 𝐺 ∈ Cℋ & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ ℋ → (0 ≤ ((((projℎ‘𝐺)‘𝐴) −ℎ ((projℎ‘𝐻)‘𝐴)) ·ih 𝐴) ↔ (normℎ‘((projℎ‘𝐻)‘𝐴)) ≤ (normℎ‘((projℎ‘𝐺)‘𝐴)))) |
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