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Type | Label | Description |
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Statement | ||
Theorem | ajval 27101* | Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑄 = (·𝑖OLD‘𝑊) & ⊢ 𝐴 = (𝑈adj𝑊) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝐴‘𝑇) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))) | ||
Syntax | ccbn 27102 | Extend class notation with the class of all complex Banach spaces. |
class CBan | ||
Definition | df-cbn 27103 | Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.) |
⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))} | ||
Theorem | iscbn 27104 | A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋))) | ||
Theorem | cbncms 27105 | The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋)) | ||
Theorem | bnnv 27106 | Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec) | ||
Theorem | bnrel 27107 | The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
⊢ Rel CBan | ||
Theorem | bnsscmcl 27108 | A subspace of a Banach space is a Banach space iff it is closed in the norm-induced metric of the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐻 = (SubSp‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) ⇒ ⊢ ((𝑈 ∈ CBan ∧ 𝑊 ∈ 𝐻) → (𝑊 ∈ CBan ↔ 𝑌 ∈ (Clsd‘𝐽))) | ||
Theorem | cnbn 27109 | The set of complex numbers is a complex Banach space. (Contributed by Steve Rodriguez, 4-Jan-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 + , · 〉, abs〉 ⇒ ⊢ 𝑈 ∈ CBan | ||
Theorem | ubthlem1 27110* | Lemma for ubth 27113. The function 𝐴 exhibits a countable collection of sets that are closed, being the inverse image under 𝑡 of the closed ball of radius 𝑘, and by assumption they cover 𝑋. Thus, by the Baire Category theorem bcth2 22935, for some 𝑛 the set 𝐴‘𝑛 has an interior, meaning that there is a closed ball {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} in the set. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑊) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑈 ∈ CBan & ⊢ 𝑊 ∈ NrmCVec & ⊢ (𝜑 → 𝑇 ⊆ (𝑈 BLnOp 𝑊)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐) & ⊢ 𝐴 = (𝑘 ∈ ℕ ↦ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘}) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)) | ||
Theorem | ubthlem2 27111* | Lemma for ubth 27113. Given that there is a closed ball 𝐵(𝑃, 𝑅) in 𝐴‘𝐾, for any 𝑥 ∈ 𝐵(0, 1), we have 𝑃 + 𝑅 · 𝑥 ∈ 𝐵(𝑃, 𝑅) and 𝑃 ∈ 𝐵(𝑃, 𝑅), so both of these have norm(𝑡(𝑧)) ≤ 𝐾 and so norm(𝑡(𝑥 )) ≤ (norm(𝑡(𝑃)) + norm(𝑡(𝑃 + 𝑅 · 𝑥))) / 𝑅 ≤ ( 𝐾 + 𝐾) / 𝑅, which is our desired uniform bound. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑊) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑈 ∈ CBan & ⊢ 𝑊 ∈ NrmCVec & ⊢ (𝜑 → 𝑇 ⊆ (𝑈 BLnOp 𝑊)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐) & ⊢ 𝐴 = (𝑘 ∈ ℕ ↦ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘}) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ⊆ (𝐴‘𝐾)) ⇒ ⊢ (𝜑 → ∃𝑑 ∈ ℝ ∀𝑡 ∈ 𝑇 ((𝑈 normOpOLD 𝑊)‘𝑡) ≤ 𝑑) | ||
Theorem | ubthlem3 27112* | Lemma for ubth 27113. Prove the reverse implication, using nmblolbi 27039. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑊) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑈 ∈ CBan & ⊢ 𝑊 ∈ NrmCVec & ⊢ (𝜑 → 𝑇 ⊆ (𝑈 BLnOp 𝑊)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 ↔ ∃𝑑 ∈ ℝ ∀𝑡 ∈ 𝑇 ((𝑈 normOpOLD 𝑊)‘𝑡) ≤ 𝑑)) | ||
Theorem | ubth 27113* | Uniform Boundedness Theorem, also called the Banach-Steinhaus Theorem. Let 𝑇 be a collection of bounded linear operators on a Banach space. If, for every vector 𝑥, the norms of the operators' values are bounded, then the operators' norms are also bounded. Theorem 4.7-3 of [Kreyszig] p. 249. See also http://en.wikipedia.org/wiki/Uniform_boundedness_principle. (Contributed by NM, 7-Nov-2007.) (Proof shortened by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑊) & ⊢ 𝑀 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ ((𝑈 ∈ CBan ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ⊆ (𝑈 BLnOp 𝑊)) → (∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 ↔ ∃𝑑 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑀‘𝑡) ≤ 𝑑)) | ||
Theorem | minvecolem1 27114* | Lemma for minveco 27124. The set of all distances from points of 𝑌 to 𝐴 are a nonempty set of nonnegative reals. (Contributed by Mario Carneiro, 8-May-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ⇒ ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) | ||
Theorem | minvecolem2 27115* | Lemma for minveco 27124. Any two points 𝐾 and 𝐿 in 𝑌 are close to each other if they are close to the infimum of distance to 𝐴. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐵) & ⊢ (𝜑 → 𝐾 ∈ 𝑌) & ⊢ (𝜑 → 𝐿 ∈ 𝑌) & ⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵)) & ⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵)) ⇒ ⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵)) | ||
Theorem | minvecolem3 27116* | Lemma for minveco 27124. The sequence formed by taking elements successively closer to the infimum is Cauchy. (Contributed by Mario Carneiro, 8-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑌) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))) ⇒ ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) | ||
Theorem | minvecolem4a 27117* | Lemma for minveco 27124. 𝐹 is convergent in the subspace topology on 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑌) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))) ⇒ ⊢ (𝜑 → 𝐹(⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹)) | ||
Theorem | minvecolem4b 27118* | Lemma for minveco 27124. The convergent point of the cauchy sequence 𝐹 is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑌) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))) ⇒ ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘𝐹) ∈ 𝑋) | ||
Theorem | minvecolem4c 27119* | Lemma for minveco 27124. The infimum of the distances to 𝐴 is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑌) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))) ⇒ ⊢ (𝜑 → 𝑆 ∈ ℝ) | ||
Theorem | minvecolem4 27120* | Lemma for minveco 27124. The convergent point of the cauchy sequence 𝐹 attains the minimum distance, and so is closer to 𝐴 than any other point in 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑌) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))) & ⊢ 𝑇 = (1 / (((((𝐴𝐷((⇝𝑡‘𝐽)‘𝐹)) + 𝑆) / 2)↑2) − (𝑆↑2))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦))) | ||
Theorem | minvecolem5 27121* | Lemma for minveco 27124. Discharge the assumption about the sequence 𝐹 by applying countable choice ax-cc 9140. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦))) | ||
Theorem | minvecolem6 27122* | Lemma for minveco 27124. Any minimal point is less than 𝑆 away from 𝐴. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (((𝐴𝐷𝑥)↑2) ≤ ((𝑆↑2) + 0) ↔ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))) | ||
Theorem | minvecolem7 27123* | Lemma for minveco 27124. Since any two minimal points are distance zero away from each other, the minimal point is unique. (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦))) | ||
Theorem | minveco 27124* | Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace 𝑊 that minimizes the distance to an arbitrary vector 𝐴 in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Mario Carneiro, 9-May-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦))) | ||
Syntax | chlo 27125 | Extend class notation with the class of all complex Hilbert spaces. |
class CHilOLD | ||
Definition | df-hlo 27126 | Define the class of all complex Hilbert spaces. A Hilbert space is a Banach space which is also an inner product space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.) |
⊢ CHilOLD = (CBan ∩ CPreHilOLD) | ||
Theorem | ishlo 27127 | The predicate "is a complex Hilbert space." A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.) |
⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD)) | ||
Theorem | hlobn 27128 | Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.) |
⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan) | ||
Theorem | hlph 27129 | Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (New usage is discouraged.) |
⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD) | ||
Theorem | hlrel 27130 | The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
⊢ Rel CHilOLD | ||
Theorem | hlnv 27131 | Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) | ||
Theorem | hlnvi 27132 | Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.) |
⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ 𝑈 ∈ NrmCVec | ||
Theorem | hlvc 27133 | Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑊 = (1st ‘𝑈) ⇒ ⊢ (𝑈 ∈ CHilOLD → 𝑊 ∈ CVecOLD) | ||
Theorem | hlcmet 27134 | The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋)) | ||
Theorem | hlmet 27135 | The induced metric on a complex Hilbert space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (Met‘𝑋)) | ||
Theorem | hlpar2 27136 | The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | ||
Theorem | hlpar 27137 | The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | ||
Theorem | hlex 27138 | The base set of a Hilbert space is a set. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) ⇒ ⊢ 𝑋 ∈ V | ||
Theorem | hladdf 27139 | Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) ⇒ ⊢ (𝑈 ∈ CHilOLD → 𝐺:(𝑋 × 𝑋)⟶𝑋) | ||
Theorem | hlcom 27140 | Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)) | ||
Theorem | hlass 27141 | Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶))) | ||
Theorem | hl0cl 27142 | The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) ⇒ ⊢ (𝑈 ∈ CHilOLD → 𝑍 ∈ 𝑋) | ||
Theorem | hladdid 27143 | Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) | ||
Theorem | hlmulf 27144 | Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ (𝑈 ∈ CHilOLD → 𝑆:(ℂ × 𝑋)⟶𝑋) | ||
Theorem | hlmulid 27145 | Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) | ||
Theorem | hlmulass 27146 | Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶))) | ||
Theorem | hldi 27147 | Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶))) | ||
Theorem | hldir 27148 | Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))) | ||
Theorem | hlmul0 27149 | Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍) | ||
Theorem | hlipf 27150 | Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ (𝑈 ∈ CHilOLD → 𝑃:(𝑋 × 𝑋)⟶ℂ) | ||
Theorem | hlipcj 27151 | Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (∗‘(𝐵𝑃𝐴))) | ||
Theorem | hlipdir 27152 | Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶))) | ||
Theorem | hlipass 27153 | Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))) | ||
Theorem | hlipgt0 27154 | The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → 0 < (𝐴𝑃𝐴)) | ||
Theorem | hlcompl 27155 | Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.) |
⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ dom (⇝𝑡‘𝐽)) | ||
Theorem | cnchl 27156 | The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 + , · 〉, abs〉 ⇒ ⊢ 𝑈 ∈ CHilOLD | ||
Theorem | ssphl 27157 | A Banach subspace of an inner product space is a Hilbert space. (Contributed by NM, 11-Apr-2008.) (New usage is discouraged.) |
⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ 𝐻 ∧ 𝑊 ∈ CBan) → 𝑊 ∈ CHilOLD) | ||
Theorem | htthlem 27158* | Lemma for htth 27159. The collection 𝐾, which consists of functions 𝐹(𝑧)(𝑤) = 〈𝑤 ∣ 𝑇(𝑧)〉 = 〈𝑇(𝑤) ∣ 𝑧〉 for each 𝑧 in the unit ball, is a collection of bounded linear functions by ipblnfi 27095, so by the Uniform Boundedness theorem ubth 27113, there is a uniform bound 𝑦 on ∥ 𝐹(𝑥) ∥ for all 𝑥 in the unit ball. Then ∣ 𝑇(𝑥) ∣ ↑2 = 〈𝑇(𝑥) ∣ 𝑇(𝑥)〉 = 𝐹(𝑥)( 𝑇(𝑥)) ≤ 𝑦 ∣ 𝑇(𝑥) ∣, so ∣ 𝑇(𝑥) ∣ ≤ 𝑦 and 𝑇 is bounded. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝐿 = (𝑈 LnOp 𝑈) & ⊢ 𝐵 = (𝑈 BLnOp 𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑈 ∈ CHilOLD & ⊢ 𝑊 = 〈〈 + , · 〉, abs〉 & ⊢ (𝜑 → 𝑇 ∈ 𝐿) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) & ⊢ 𝐹 = (𝑧 ∈ 𝑋 ↦ (𝑤 ∈ 𝑋 ↦ (𝑤𝑃(𝑇‘𝑧)))) & ⊢ 𝐾 = (𝐹 “ {𝑧 ∈ 𝑋 ∣ (𝑁‘𝑧) ≤ 1}) ⇒ ⊢ (𝜑 → 𝑇 ∈ 𝐵) | ||
Theorem | htth 27159* | Hellinger-Toeplitz Theorem: any self-adjoint linear operator defined on all of Hilbert space is bounded. Theorem 10.1-1 of [Kreyszig] p. 525. Discovered by E. Hellinger and O. Toeplitz in 1910, "it aroused both admiration and puzzlement since the theorem establishes a relation between properties of two different kinds, namely, the properties of being defined everywhere and being bounded." (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝐿 = (𝑈 LnOp 𝑈) & ⊢ 𝐵 = (𝑈 BLnOp 𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ 𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) → 𝑇 ∈ 𝐵) | ||
This part contains the definitions and theorems used by the Hilbert Space Explorer (HSE), mmhil.html. Because it axiomatizes a single complex Hilbert space whose existence is assumed, its usefulness is limited. For example, it cannot work with real or quaternion Hilbert spaces and it cannot study relationships between two Hilbert spaces. More information can be found on the Hilbert Space Explorer page. Future development should instead work with general Hilbert spaces as defined by df-hil 19867; note that df-hil 19867 uses extensible structures. The intent is for this deprecated section to be deleted once all its theorems have been translated into extensible structure versions (or are not useful). Many of the theorems in this section have already been translated to extensible structure versions, but there is still a lot in this section that might be useful for future reference. It is much easier to translate these by hand from this section than to start from scratch from textbook proofs, since the HSE omits no details. | ||
Syntax | chil 27160 | Extend class notation with Hilbert vector space. |
class ℋ | ||
Syntax | cva 27161 | Extend class notation with vector addition in Hilbert space. In the literature, the subscript "v" is omitted, but we need it to avoid ambiguity with complex number addition + caddc 9818. |
class +ℎ | ||
Syntax | csm 27162 | Extend class notation with scalar multiplication in Hilbert space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity. |
class ·ℎ | ||
Syntax | csp 27163 | Extend class notation with inner (scalar) product in Hilbert space. In the literature, the inner product of 𝐴 and 𝐵 is usually written 〈𝐴, 𝐵〉 but our operation notation allows us to use existing theorems about operations and also eliminates ambiguity with the definition of an ordered pair df-op 4132. |
class ·ih | ||
Syntax | cno 27164 | Extend class notation with the norm function in Hilbert space. In the literature, the norm of 𝐴 is usually written "|| 𝐴 ||", but we use function notation to take advantage of our existing theorems about functions. |
class normℎ | ||
Syntax | c0v 27165 | Extend class notation with zero vector in Hilbert space. |
class 0ℎ | ||
Syntax | cmv 27166 | Extend class notation with vector subtraction in Hilbert space. |
class −ℎ | ||
Syntax | ccau 27167 | Extend class notation with set of Cauchy sequences in Hilbert space. |
class Cauchy | ||
Syntax | chli 27168 | Extend class notation with convergence relation in Hilbert space. |
class ⇝𝑣 | ||
Syntax | csh 27169 | Extend class notation with set of subspaces of a Hilbert space. |
class Sℋ | ||
Syntax | cch 27170 | Extend class notation with set of closed subspaces of a Hilbert space. |
class Cℋ | ||
Syntax | cort 27171 | Extend class notation with orthogonal complement in Cℋ. |
class ⊥ | ||
Syntax | cph 27172 | Extend class notation with subspace sum in Cℋ. |
class +ℋ | ||
Syntax | cspn 27173 | Extend class notation with subspace span in Cℋ. |
class span | ||
Syntax | chj 27174 | Extend class notation with join in Cℋ. |
class ∨ℋ | ||
Syntax | chsup 27175 | Extend class notation with supremum of a collection in Cℋ. |
class ∨ℋ | ||
Syntax | c0h 27176 | Extend class notation with zero of Cℋ. |
class 0ℋ | ||
Syntax | ccm 27177 | Extend class notation with the commutes relation on a Hilbert lattice. |
class 𝐶ℋ | ||
Syntax | cpjh 27178 | Extend class notation with set of projections on a Hilbert space. |
class projℎ | ||
Syntax | chos 27179 | Extend class notation with sum of Hilbert space operators. |
class +op | ||
Syntax | chot 27180 | Extend class notation with scalar product of a Hilbert space operator. |
class ·op | ||
Syntax | chod 27181 | Extend class notation with difference of Hilbert space operators. |
class −op | ||
Syntax | chfs 27182 | Extend class notation with sum of Hilbert space functionals. |
class +fn | ||
Syntax | chft 27183 | Extend class notation with scalar product of Hilbert space functional. |
class ·fn | ||
Syntax | ch0o 27184 | Extend class notation with the Hilbert space zero operator. |
class 0hop | ||
Syntax | chio 27185 | Extend class notation with Hilbert space identity operator. |
class Iop | ||
Syntax | cnop 27186 | Extend class notation with the operator norm function. |
class normop | ||
Syntax | ccop 27187 | Extend class notation with set of continuous Hilbert space operators. |
class ConOp | ||
Syntax | clo 27188 | Extend class notation with set of linear Hilbert space operators. |
class LinOp | ||
Syntax | cbo 27189 | Extend class notation with set of bounded linear operators. |
class BndLinOp | ||
Syntax | cuo 27190 | Extend class notation with set of unitary Hilbert space operators. |
class UniOp | ||
Syntax | cho 27191 | Extend class notation with set of Hermitian Hilbert space operators. |
class HrmOp | ||
Syntax | cnmf 27192 | Extend class notation with the functional norm function. |
class normfn | ||
Syntax | cnl 27193 | Extend class notation with the functional nullspace function. |
class null | ||
Syntax | ccnfn 27194 | Extend class notation with set of continuous Hilbert space functionals. |
class ConFn | ||
Syntax | clf 27195 | Extend class notation with set of linear Hilbert space functionals. |
class LinFn | ||
Syntax | cado 27196 | Extend class notation with Hilbert space adjoint function. |
class adjℎ | ||
Syntax | cbr 27197 | Extend class notation with the bra of a vector in Dirac bra-ket notation. |
class bra | ||
Syntax | ck 27198 | Extend class notation with the outer product of two vectors in Dirac bra-ket notation. |
class ketbra | ||
Syntax | cleo 27199 | Extend class notation with positive operator ordering. |
class ≤op | ||
Syntax | cei 27200 | Extend class notation with Hilbert space eigenvector function. |
class eigvec |
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