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Theorem List for Metamath Proof Explorer - 30401-30500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnmcoplb 30401 A lower bound for the norm of a continuous linear Hilbert space operator. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
((𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp ∧ 𝐴 ∈ ℋ) → (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴)))
 
Theoremnmophmi 30402 The norm of the scalar product of a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑇 ∈ BndLinOp       (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) = ((abs‘𝐴) · (normop𝑇)))
 
Theorembdophmi 30403 The scalar product of a bounded linear operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑇 ∈ BndLinOp       (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ BndLinOp)
 
Theoremlnconi 30404* Lemma for lnopconi 30405 and lnfnconi 30426. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
(𝑇𝐶𝑆 ∈ ℝ)    &   ((𝑇𝐶𝑦 ∈ ℋ) → (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦)))    &   (𝑇𝐶 ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+𝑦 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧))    &   (𝑦 ∈ ℋ → (𝑁‘(𝑇𝑦)) ∈ ℝ)    &   ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘(𝑤 𝑥)) = ((𝑇𝑤)𝑀(𝑇𝑥)))       (𝑇𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))
 
Theoremlnopconi 30405* A condition equivalent to "𝑇 is continuous" when 𝑇 is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
𝑇 ∈ LinOp       (𝑇 ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (norm‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))
 
Theoremlnopcon 30406* 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.) (New usage is discouraged.)
(𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (norm‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))))
 
Theoremlnopcnbd 30407 A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp))
 
Theoremlncnopbd 30408 A continuous linear operator is a bounded linear operator. This theorem justifies our use of "bounded linear" as an interchangeable condition for "continuous linear" used in some textbook proofs. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ (LinOp ∩ ContOp) ↔ 𝑇 ∈ BndLinOp)
 
Theoremlncnbd 30409 A continuous linear operator is a bounded linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
(LinOp ∩ ContOp) = BndLinOp
 
Theoremlnopcnre 30410 A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ (normop𝑇) ∈ ℝ))
 
Theoremlnfnli 30411 Basic property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
 
Theoremlnfnfi 30412 A linear Hilbert space functional is a functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn       𝑇: ℋ⟶ℂ
 
Theoremlnfn0i 30413 The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn       (𝑇‘0) = 0
 
Theoremlnfnaddi 30414 Additive property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 + 𝐵)) = ((𝑇𝐴) + (𝑇𝐵)))
 
Theoremlnfnmuli 30415 Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 · 𝐵)) = (𝐴 · (𝑇𝐵)))
 
Theoremlnfnaddmuli 30416 Sum/product property of a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 + (𝐴 · 𝐶))) = ((𝑇𝐵) + (𝐴 · (𝑇𝐶))))
 
Theoremlnfnsubi 30417 Subtraction property for a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 𝐵)) = ((𝑇𝐴) − (𝑇𝐵)))
 
Theoremlnfn0 30418 The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ LinFn → (𝑇‘0) = 0)
 
Theoremlnfnmul 30419 Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
((𝑇 ∈ LinFn ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 · 𝐵)) = (𝐴 · (𝑇𝐵)))
 
Theoremnmbdfnlbi 30420 A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ)       (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
 
Theoremnmbdfnlb 30421 A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
 
Theoremnmcfnexi 30422 The norm of a continuous linear Hilbert space functional exists. Theorem 3.5(i) 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       (normfn𝑇) ∈ ℝ
 
Theoremnmcfnlbi 30423 A lower bound for the norm of a continuous linear 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)))
 
Theoremnmcfnex 30424 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 30425 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 30426* 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 30427* 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 30428 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 30429 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 30430 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 30431 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 30432 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 30433* 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 30434* 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 30435* A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. See riesz2 30437 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 30436* 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 30437. For the continuous linear functional version, see riesz3i 30433 and riesz4 30435. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ LinFn → ((normfn𝑇) ∈ ℝ ↔ ∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑥 ·ih 𝑦)))
 
Theoremriesz2 30437* 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 30436. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ) → ∃!𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑥 ·ih 𝑦))
 
19.6.12  Adjoints (cont.)
 
Theoremcnlnadjlem1 30438* Lemma for cnlnadji 30447 (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 30439* Lemma for cnlnadji 30447. 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 30440* Lemma for cnlnadji 30447. By riesz4 30435, 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 30441* Lemma for cnlnadji 30447. 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 30442* Lemma for cnlnadji 30447. 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 30443* Lemma for cnlnadji 30447. 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 30444* Lemma for cnlnadji 30447. 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 30445* Lemma for cnlnadji 30447. 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 30446* Lemma for cnlnadji 30447. 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 30447* 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 30448* 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 30449* 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 30450* 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 30451 Every continuous linear Hilbert space operator has an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
(LinOp ∩ ContOp) ⊆ dom adj
 
Theorembdopssadj 30452 Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
BndLinOp ⊆ dom adj
 
Theorembdopadj 30453 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 30454 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 30455 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 30456 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 30457 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 30458 Every operator with an adjoint is linear. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)
dom adj ⊆ LinOp
 
Theoremnmopadjlei 30459 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 30460 Lemma for nmopadji 30461. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ BndLinOp       (normop‘(adj𝑇)) ≤ (normop𝑇)
 
Theoremnmopadji 30461 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 30462 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 30463 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 30464 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 30465 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 30466 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 30467 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 30468 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 30469 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 30470 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 30471 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 30472 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 30473 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 30474 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 30475 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 30476 The norm of the bra function. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (norm𝐴))
 
Theorembrabn 30477 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 30478 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 30479 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 30480 A bra is a continuous linear functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (bra‘𝐴) ∈ (LinFn ∩ ContFn))
 
Theoremcnvbraval 30481* Value of the converse of the bra function. Based on the Riesz Lemma riesz4 30435, 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 30482 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 30483 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 30484 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 30485* 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 30486 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 30487 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 30488 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 30489 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 30490 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 30491 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 30492 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 30493* 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 30494* 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 30495* 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 30496 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 30497* 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 30498 The ordering relation for operators is reflexive. (Contributed by NM, 24-Jul-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → 𝑇op 𝑇)
 
Theoremleoprf 30499 The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇op 𝑇)
 
Theoremleopsq 30500 The square of a Hermitian operator is positive. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 0hopop (𝑇𝑇))
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