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Type | Label | Description |
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Statement | ||
Theorem | nmcfnex 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) → (normfn‘𝑇) ∈ ℝ) | ||
Theorem | nmcfnlb 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‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))) | ||
Theorem | lnfnconi 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‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) | ||
Theorem | lnfncon 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‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))) | ||
Theorem | lnfncnbd 29505 | A linear functional is continuous iff it is bounded. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ LinFn → (𝑇 ∈ ContFn ↔ (normfn‘𝑇) ∈ ℝ)) | ||
Theorem | imaelshi 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 & ⊢ 𝐴 ∈ Sℋ ⇒ ⊢ (𝑇 “ 𝐴) ∈ Sℋ | ||
Theorem | rnelshi 29507 | The range of a linear operator is a subspace. (Contributed by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
⊢ 𝑇 ∈ LinOp ⇒ ⊢ ran 𝑇 ∈ Sℋ | ||
Theorem | nlelshi 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) ∈ Sℋ | ||
Theorem | nlelchi 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) ∈ Cℋ | ||
Theorem | riesz3i 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 (𝑇‘𝑣) = (𝑣 ·ih 𝑤) | ||
Theorem | riesz4i 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 (𝑇‘𝑣) = (𝑣 ·ih 𝑤) | ||
Theorem | riesz4 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 (𝑇‘𝑣) = (𝑣 ·ih 𝑤)) | ||
Theorem | riesz1 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 → ((normfn‘𝑇) ∈ ℝ ↔ ∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦))) | ||
Theorem | riesz2 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 ∧ (normfn‘𝑇) ∈ ℝ) → ∃!𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦)) | ||
Theorem | cnlnadjlem1 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 ↦ ((𝑇‘𝑔) ·ih 𝑦)) ⇒ ⊢ (𝐴 ∈ ℋ → (𝐺‘𝐴) = ((𝑇‘𝐴) ·ih 𝑦)) | ||
Theorem | cnlnadjlem2 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 ↦ ((𝑇‘𝑔) ·ih 𝑦)) ⇒ ⊢ (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn)) | ||
Theorem | cnlnadjlem3 29517* | Lemma for cnlnadji 29524. By riesz4 29512, 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 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 ↦ ((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 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 ↦ ((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 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 ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ 𝐹 ∈ LinOp | ||
Theorem | cnlnadjlem7 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 ↦ ((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 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 ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ 𝐹 ∈ ContOp | ||
Theorem | cnlnadjlem9 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 ↦ ((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 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 ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦)) | ||
Theorem | cnlnadjeui 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 ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦)) | ||
Theorem | cnlnadjeu 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 ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦))) | ||
Theorem | cnlnadj 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 ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦))) | ||
Theorem | cnlnssadj 29528 | 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 29529 | Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
⊢ BndLinOp ⊆ dom adjℎ | ||
Theorem | bdopadj 29530 | 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 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 → (adjℎ‘𝑇) ∈ BndLinOp) | ||
Theorem | adjbdlnb 29532 | 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 29533 | 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 29534 | 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 29535 | Every operator with an adjoint is linear. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.) |
⊢ dom adjℎ ⊆ LinOp | ||
Theorem | nmopadjlei 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 → (normℎ‘((adjℎ‘𝑇)‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
Theorem | nmopadjlem 29537 | Lemma for nmopadji 29538. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇) | ||
Theorem | nmopadji 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‘(adjℎ‘𝑇)) = (normop‘𝑇) | ||
Theorem | adjeq0 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 ↔ (adjℎ‘𝑇) = 0hop ) | ||
Theorem | adjmul 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.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝐴 ·op 𝑇)) = ((∗‘𝐴) ·op (adjℎ‘𝑇))) | ||
Theorem | adjadd 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.) |
⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝑆 +op 𝑇)) = ((adjℎ‘𝑆) +op (adjℎ‘𝑇))) | ||
Theorem | nmoptrii 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)) ≤ ((normop‘𝑆) + (normop‘𝑇)) | ||
Theorem | nmopcoi 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‘(𝑆 ∘ 𝑇)) ≤ ((normop‘𝑆) · (normop‘𝑇)) | ||
Theorem | bdophsi 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 | ||
Theorem | bdophdi 29545 | 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 29546 | 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 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 ⇒ ⊢ ((normop‘𝑆) − (normop‘𝑇)) ≤ (normop‘(𝑆 +op 𝑇)) | ||
Theorem | adjcoi 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.) |
⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (adjℎ‘(𝑆 ∘ 𝑇)) = ((adjℎ‘𝑇) ∘ (adjℎ‘𝑆)) | ||
Theorem | nmopcoadji 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‘((adjℎ‘𝑇) ∘ 𝑇)) = ((normop‘𝑇)↑2) | ||
Theorem | nmopcoadj2i 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 ∘ (adjℎ‘𝑇))) = ((normop‘𝑇)↑2) | ||
Theorem | nmopcoadj0i 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 ∘ (adjℎ‘𝑇)) = 0hop ↔ 𝑇 = 0hop ) | ||
Theorem | unierri 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‘((𝐹 ∘ 𝐺) −op (𝑆 ∘ 𝑇))) ≤ ((normop‘(𝐹 −op 𝑆)) + (normop‘(𝐺 −op 𝑇))) | ||
Theorem | branmfn 29553 | 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 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) | ||
Theorem | rnbra 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) | ||
Theorem | bra11 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) | ||
Theorem | bracnln 29557 | 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 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 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 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) | ||
Theorem | cnvbrabra 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) | ||
Theorem | bracnvbra 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) | ||
Theorem | bracnlnval 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 (𝑇‘𝑥) = (𝑥 ·ih 𝑦)))) | ||
Theorem | cnvbramul 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) ·ℎ (◡bra‘𝑇))) | ||
Theorem | kbass1 29564 | 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 29565 | 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 29566 | 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 29567 | 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 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)) | ||
Theorem | kbass6 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))))) | ||
Theorem | leopg 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.) |
⊢ ((𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐵) → (𝑇 ≤op 𝑈 ↔ ((𝑈 −op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥)))) | ||
Theorem | leop 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) → (𝑇 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥))) | ||
Theorem | leop2 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) → (𝑇 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) | ||
Theorem | leop3 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) → (𝑇 ≤op 𝑈 ↔ 0hop ≤op (𝑈 −op 𝑇))) | ||
Theorem | leoppos 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 → ( 0hop ≤op 𝑇 ↔ ∀𝑥 ∈ ℋ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥))) | ||
Theorem | leoprf2 29575 | 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 29576 | The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ HrmOp → 𝑇 ≤op 𝑇) | ||
Theorem | leopsq 29577 | 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 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.) |
⊢ 0hop ≤op 0hop | ||
Theorem | idleop 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.) |
⊢ 0hop ≤op Iop | ||
Theorem | leopadd 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) ∧ ( 0hop ≤op 𝑇 ∧ 0hop ≤op 𝑈)) → 0hop ≤op (𝑇 +op 𝑈)) | ||
Theorem | leopmuli 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 ∧ 0hop ≤op 𝑇)) → 0hop ≤op (𝐴 ·op 𝑇)) | ||
Theorem | leopmul 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) → ( 0hop ≤op 𝑇 ↔ 0hop ≤op (𝐴 ·op 𝑇))) | ||
Theorem | leopmul2i 29583 | 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 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) → ((𝑇 ≤op 𝑈 ∧ 𝑈 ≤op 𝑇) ↔ 𝑇 = 𝑈)) | ||
Theorem | leoptr 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) ∧ (𝑆 ≤op 𝑇 ∧ 𝑇 ≤op 𝑈)) → 𝑆 ≤op 𝑈) | ||
Theorem | leopnmid 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 ∧ (normop‘𝑇) ∈ ℝ) → 𝑇 ≤op ((normop‘𝑇) ·op Iop )) | ||
Theorem | nmopleid 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 ∧ (normop‘𝑇) ∈ ℝ ∧ 𝑇 ≠ 0hop ) → ((1 / (normop‘𝑇)) ·op 𝑇) ≤op Iop ) | ||
Theorem | opsqrlem1 29588* | 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 29589* | 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 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 (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 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 (𝑇 −op (𝑥 ∘ 𝑥))))) & ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) ⇒ ⊢ 𝐹:ℕ⟶HrmOp | ||
Theorem | opsqrlem5 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 (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 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 (𝑇 −op (𝑥 ∘ 𝑥))))) & ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) & ⊢ 𝑇 ≤op Iop ⇒ ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤op Iop ) | ||
Theorem | pjhmopi 29594 | A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻) ∈ HrmOp | ||
Theorem | pjlnopi 29595 | A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻) ∈ LinOp | ||
Theorem | pjnmopi 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.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐻 ≠ 0ℋ → (normop‘(projℎ‘𝐻)) = 1) | ||
Theorem | pjbdlni 29597 | A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (projℎ‘𝐻) ∈ BndLinOp | ||
Theorem | pjhmop 29598 | A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) ∈ HrmOp) | ||
Theorem | hmopidmchi 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 & ⊢ (𝑇 ∘ 𝑇) = 𝑇 ⇒ ⊢ ran 𝑇 ∈ Cℋ | ||
Theorem | hmopidmpji 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 & ⊢ (𝑇 ∘ 𝑇) = 𝑇 ⇒ ⊢ 𝑇 = (projℎ‘ran 𝑇) |
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