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Theorem List for Metamath Proof Explorer - 29501-29600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnmcfnex 29501 The norm of a continuous linear Hilbert space functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
((𝑇 ∈ LinFn ∧ 𝑇 ∈ ContFn) → (normfn𝑇) ∈ ℝ)

Theoremnmcfnlb 29502 A lower bound of the norm of a continuous linear Hilbert space functional. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
((𝑇 ∈ LinFn ∧ 𝑇 ∈ ContFn ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))

Theoremlnfnconi 29503* A condition equivalent to "𝑇 is continuous" when 𝑇 is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
𝑇 ∈ LinFn       (𝑇 ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))

Theoremlnfncon 29504* A condition equivalent to "𝑇 is continuous" when 𝑇 is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ LinFn → (𝑇 ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))))

Theoremlnfncnbd 29505 A linear functional is continuous iff it is bounded. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ LinFn → (𝑇 ∈ ContFn ↔ (normfn𝑇) ∈ ℝ))

Theoremimaelshi 29506 The image of a subspace under a linear operator is a subspace. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝐴S       (𝑇𝐴) ∈ S

Theoremrnelshi 29507 The range of a linear operator is a subspace. (Contributed by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
𝑇 ∈ LinOp       ran 𝑇S

Theoremnlelshi 29508 The null space of a linear functional is a subspace. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
𝑇 ∈ LinFn       (null‘𝑇) ∈ S

Theoremnlelchi 29509 The null space of a continuous linear functional is a closed subspace. Remark 3.8 of [Beran] p. 103. (Contributed by NM, 11-Feb-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝑇 ∈ LinFn    &   𝑇 ∈ ContFn       (null‘𝑇) ∈ C

19.6.11  Riesz lemma

Theoremriesz3i 29510* A continuous linear functional can be expressed as an inner product. Existence part of Theorem 3.9 of [Beran] p. 104. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn    &   𝑇 ∈ ContFn       𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇𝑣) = (𝑣 ·ih 𝑤)

Theoremriesz4i 29511* A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn    &   𝑇 ∈ ContFn       ∃!𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇𝑣) = (𝑣 ·ih 𝑤)

Theoremriesz4 29512* A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. See riesz2 29514 for the bounded linear functional version. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ (LinFn ∩ ContFn) → ∃!𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇𝑣) = (𝑣 ·ih 𝑤))

Theoremriesz1 29513* Part 1 of the Riesz representation theorem for bounded linear functionals. A linear functional is bounded iff its value can be expressed as an inner product. Part of Theorem 17.3 of [Halmos] p. 31. For part 2, see riesz2 29514. For the continuous linear functional version, see riesz3i 29510 and riesz4 29512. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ LinFn → ((normfn𝑇) ∈ ℝ ↔ ∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑥 ·ih 𝑦)))

Theoremriesz2 29514* Part 2 of the Riesz representation theorem for bounded linear functionals. The value of a bounded linear functional corresponds to a unique inner product. Part of Theorem 17.3 of [Halmos] p. 31. For part 1, see riesz1 29513. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ) → ∃!𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑥 ·ih 𝑦))

Theoremcnlnadjlem1 29515* Lemma for cnlnadji 29524 (Theorem 3.10 of [Beran] p. 104: every continuous linear operator has an adjoint). The value of the auxiliary functional 𝐺. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑇 ∈ ContOp    &   𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))       (𝐴 ∈ ℋ → (𝐺𝐴) = ((𝑇𝐴) ·ih 𝑦))

Theoremcnlnadjlem2 29516* Lemma for cnlnadji 29524. 𝐺 is a continuous linear functional. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑇 ∈ ContOp    &   𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))       (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))

Theoremcnlnadjlem3 29517* Lemma for cnlnadji 29524. By riesz4 29512, 𝐵 is the unique vector such that (𝑇𝑣) ·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 ↦ ((𝑇𝑔) ·ih 𝑦))    &   𝐵 = (𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))       (𝑦 ∈ ℋ → 𝐵 ∈ ℋ)

Theoremcnlnadjlem4 29518* Lemma for cnlnadji 29524. 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 ↦ ((𝑇𝑔) ·ih 𝑦))    &   𝐵 = (𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)       (𝐴 ∈ ℋ → (𝐹𝐴) ∈ ℋ)

Theoremcnlnadjlem5 29519* Lemma for cnlnadji 29524. 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 ↦ ((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 29520* Lemma for cnlnadji 29524. 𝐹 is linear. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑇 ∈ ContOp    &   𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))    &   𝐵 = (𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)       𝐹 ∈ LinOp

Theoremcnlnadjlem7 29521* Lemma for cnlnadji 29524. 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 ↦ ((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 29522* Lemma for cnlnadji 29524. 𝐹 is continuous. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑇 ∈ ContOp    &   𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))    &   𝐵 = (𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)       𝐹 ∈ ContOp

Theoremcnlnadjlem9 29523* Lemma for cnlnadji 29524. 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 ↦ ((𝑇𝑔) ·ih 𝑦))    &   𝐵 = (𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)       𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑇𝑥) ·ih 𝑧) = (𝑥 ·ih (𝑡𝑧))

Theoremcnlnadji 29524* 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 ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡𝑦))

Theoremcnlnadjeui 29525* 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 ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡𝑦))

Theoremcnlnadjeu 29526* 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 ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡𝑦)))

Theoremcnlnadj 29527* 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 ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡𝑦)))

Theoremcnlnssadj 29528 Every continuous linear Hilbert space operator has an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
(LinOp ∩ ContOp) ⊆ dom adj

Theorembdopssadj 29529 Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)

Theorembdopadj 29530 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 29531 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 29532 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 29533 The mapping of adjoints of bounded linear operators is one-to-one onto. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)

Theoremadjlnop 29534 The adjoint of an operator is linear. Proposition 1 of [AkhiezerGlazman] p. 80. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)

Theoremadjsslnop 29535 Every operator with an adjoint is linear. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)

Theoremnmopadjlei 29536 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 29537 Lemma for nmopadji 29538. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ BndLinOp       (normop‘(adj𝑇)) ≤ (normop𝑇)

Theoremnmopadji 29538 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 29539 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 29540 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.)

Theoremadjadd 29541 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.)

Theoremnmoptrii 29542 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 29543 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 29544 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 29545 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 29546 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 29547 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 29548 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.)

Theoremnmopcoadji 29549 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 29550 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 29551 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 29552 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 29553 The norm of the bra function. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (norm𝐴))

Theorembrabn 29554 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 29555 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)

Theorembra11 29556 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)

Theorembracnln 29557 A bra is a continuous linear functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (bra‘𝐴) ∈ (LinFn ∩ ContFn))

Theoremcnvbraval 29558* Value of the converse of the bra function. Based on the Riesz Lemma riesz4 29512, 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 ∩ ContFn) → (bra‘𝑇) = (𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑥 ·ih 𝑦)))

Theoremcnvbracl 29559 Closure of the converse of the bra function. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
(𝑇 ∈ (LinFn ∩ ContFn) → (bra‘𝑇) ∈ ℋ)

Theoremcnvbrabra 29560 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 29561 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)

Theorembracnlnval 29562* 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 (𝑇𝑥) = (𝑥 ·ih 𝑦))))

Theoremcnvbramul 29563 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) · (bra‘𝑇)))

Theoremkbass1 29564 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 29565 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 29566 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 29567 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 29568 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 29569 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 29570* 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 29571* 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 29572* 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 29573 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 29574* 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 29575 The ordering relation for operators is reflexive. (Contributed by NM, 24-Jul-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → 𝑇op 𝑇)

Theoremleoprf 29576 The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇op 𝑇)

Theoremleopsq 29577 The square of a Hermitian operator is positive. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 0hopop (𝑇𝑇))

Theorem0leop 29578 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 29579 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 29580 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 29581 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 29582 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 29583 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 29584 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 29585 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 29586 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 29587 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 29588* 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 29589* 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 29590* 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 29591* 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 29592* 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 29593* 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 29594 A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻) ∈ HrmOp

Theorempjlnopi 29595 A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻) ∈ LinOp

Theorempjnmopi 29596 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 29597 A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻) ∈ BndLinOp

Theorempjhmop 29598 A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → (proj𝐻) ∈ HrmOp)

Theoremhmopidmchi 29599 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 29600 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 29599 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)

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