Home Metamath Proof ExplorerTheorem List (p. 284 of 437) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28351) Hilbert Space Explorer (28352-29876) Users' Mathboxes (29877-43667)

Theorem List for Metamath Proof Explorer - 28301-28400   *Has distinct variable group(s)
TypeLabelDescription
Statement

18.6.2  Examples of complex Banach spaces

Theoremcnbn 28301 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

18.6.3  Uniform Boundedness Theorem

Theoremubthlem1 28302* Lemma for ubth 28305. 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 23540, for some 𝑛 the set 𝐴𝑛 has an interior, meaning that there is a closed ball {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} in the set. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑊)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑈 ∈ CBan    &   𝑊 ∈ NrmCVec    &   (𝜑𝑇 ⊆ (𝑈 BLnOp 𝑊))    &   (𝜑 → ∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐)    &   𝐴 = (𝑘 ∈ ℕ ↦ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})       (𝜑 → ∃𝑛 ∈ ℕ ∃𝑦𝑋𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛))

Theoremubthlem2 28303* Lemma for ubth 28305. Given that there is a closed ball 𝐵(𝑃, 𝑅) in 𝐴𝐾, for any 𝑥𝐵(0, 1), we have 𝑃 + 𝑅 · 𝑥𝐵(𝑃, 𝑅) and 𝑃𝐵(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 = (normCV𝑊)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑈 ∈ CBan    &   𝑊 ∈ NrmCVec    &   (𝜑𝑇 ⊆ (𝑈 BLnOp 𝑊))    &   (𝜑 → ∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐)    &   𝐴 = (𝑘 ∈ ℕ ↦ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑃𝑋)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → {𝑧𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ⊆ (𝐴𝐾))       (𝜑 → ∃𝑑 ∈ ℝ ∀𝑡𝑇 ((𝑈 normOpOLD 𝑊)‘𝑡) ≤ 𝑑)

Theoremubthlem3 28304* Lemma for ubth 28305. Prove the reverse implication, using nmblolbi 28231. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑊)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑈 ∈ CBan    &   𝑊 ∈ NrmCVec    &   (𝜑𝑇 ⊆ (𝑈 BLnOp 𝑊))       (𝜑 → (∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐 ↔ ∃𝑑 ∈ ℝ ∀𝑡𝑇 ((𝑈 normOpOLD 𝑊)‘𝑡) ≤ 𝑑))

Theoremubth 28305* 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 = (normCV𝑊)    &   𝑀 = (𝑈 normOpOLD 𝑊)       ((𝑈 ∈ CBan ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ⊆ (𝑈 BLnOp 𝑊)) → (∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐 ↔ ∃𝑑 ∈ ℝ ∀𝑡𝑇 (𝑀𝑡) ≤ 𝑑))

18.6.4  Minimizing Vector Theorem

Theoremminvecolem1 28306* Lemma for minveco 28316. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))       (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤𝑅 0 ≤ 𝑤))

Theoremminvecolem2 28307* Lemma for minveco 28316. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)    &   (𝜑𝐾𝑌)    &   (𝜑𝐿𝑌)    &   (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵))    &   (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵))       (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵))

Theoremminvecolem3 28308* Lemma for minveco 28316. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   (𝜑𝐹:ℕ⟶𝑌)    &   ((𝜑𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))       (𝜑𝐹 ∈ (Cau‘𝐷))

Theoremminvecolem4a 28309* Lemma for minveco 28316. 𝐹 is convergent in the subspace topology on 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   (𝜑𝐹:ℕ⟶𝑌)    &   ((𝜑𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))       (𝜑𝐹(⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹))

Theoremminvecolem4b 28310* Lemma for minveco 28316. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   (𝜑𝐹:ℕ⟶𝑌)    &   ((𝜑𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))       (𝜑 → ((⇝𝑡𝐽)‘𝐹) ∈ 𝑋)

Theoremminvecolem4c 28311* Lemma for minveco 28316. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   (𝜑𝐹:ℕ⟶𝑌)    &   ((𝜑𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))       (𝜑𝑆 ∈ ℝ)

Theoremminvecolem4 28312* Lemma for minveco 28316. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   (𝜑𝐹:ℕ⟶𝑌)    &   ((𝜑𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))    &   𝑇 = (1 / (((((𝐴𝐷((⇝𝑡𝐽)‘𝐹)) + 𝑆) / 2)↑2) − (𝑆↑2)))       (𝜑 → ∃𝑥𝑌𝑦𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

Theoremminvecolem5 28313* Lemma for minveco 28316. Discharge the assumption about the sequence 𝐹 by applying countable choice ax-cc 9594. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )       (𝜑 → ∃𝑥𝑌𝑦𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

Theoremminvecolem6 28314* Lemma for minveco 28316. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )       ((𝜑𝑥𝑌) → (((𝐴𝐷𝑥)↑2) ≤ ((𝑆↑2) + 0) ↔ ∀𝑦𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦))))

Theoremminvecolem7 28315* Lemma for minveco 28316. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )       (𝜑 → ∃!𝑥𝑌𝑦𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

Theoremminveco 28316* 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)       (𝜑 → ∃!𝑥𝑌𝑦𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

18.7  Complex Hilbert spaces

18.7.1  Definition and basic properties

Syntaxchlo 28317 Extend class notation with the class of all complex Hilbert spaces.
class CHilOLD

Definitiondf-hlo 28318 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)

Theoremishlo 28319 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))

Theoremhlobn 28320 Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
(𝑈 ∈ CHilOLD𝑈 ∈ CBan)

Theoremhlph 28321 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 ∈ CHilOLD𝑈 ∈ CPreHilOLD)

Theoremhlrel 28322 The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
Rel CHilOLD

Theoremhlnv 28323 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
(𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)

Theoremhlnvi 28324 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
𝑈 ∈ CHilOLD       𝑈 ∈ NrmCVec

Theoremhlvc 28325 Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑊 = (1st𝑈)       (𝑈 ∈ CHilOLD𝑊 ∈ CVecOLD)

Theoremhlcmet 28326 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 ∈ CHilOLD𝐷 ∈ (CMet‘𝑋))

Theoremhlmet 28327 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 ∈ CHilOLD𝐷 ∈ (Met‘𝑋))

Theoremhlpar2 28328 The parallelogram law satisfied by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋𝐵𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))

Theoremhlpar 28329 The parallelogram law satisfied by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋𝐵𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))

18.7.2  Standard axioms for a complex Hilbert space

Theoremhlex 28330 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

Theoremhladdf 28331 Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       (𝑈 ∈ CHilOLD𝐺:(𝑋 × 𝑋)⟶𝑋)

Theoremhlcom 28332 Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Theoremhlass 28333 Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ CHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))

Theoremhl0cl 28334 The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)       (𝑈 ∈ CHilOLD𝑍𝑋)

Theoremhladdid 28335 Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)

Theoremhlmulf 28336 Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       (𝑈 ∈ CHilOLD𝑆:(ℂ × 𝑋)⟶𝑋)

Theoremhlmulid 28337 Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)

Theoremhlmulass 28338 Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

Theoremhldi 28339 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))

Theoremhldir 28340 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

Theoremhlmul0 28341 Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋) → (0𝑆𝐴) = 𝑍)

Theoremhlipf 28342 Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)       (𝑈 ∈ CHilOLD𝑃:(𝑋 × 𝑋)⟶ℂ)

Theoremhlipcj 28343 Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋𝐵𝑋) → (𝐴𝑃𝐵) = (∗‘(𝐵𝑃𝐴)))

Theoremhlipdir 28344 Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Theoremhlipass 28345 Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Theoremhlipgt0 28346 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 = (0vec𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋𝐴𝑍) → 0 < (𝐴𝑃𝐴))

Theoremhlcompl 28347 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 ∈ CHilOLD𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ dom (⇝𝑡𝐽))

18.7.3  Examples of complex Hilbert spaces

Theoremcnchl 28348 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

18.7.4  Subspaces

TheoremssphlOLD 28349 Obsolete version of cphssphl 23581 as of 6-Oct-2022. A Banach subspace of an inner product space is a Hilbert space. (Contributed by NM, 11-Apr-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐻 = (SubSp‘𝑈)       ((𝑈 ∈ CPreHilOLD𝑊𝐻𝑊 ∈ CBan) → 𝑊 ∈ CHilOLD)

18.7.5  Hellinger-Toeplitz Theorem

Theoremhtthlem 28350* Lemma for htth 28351. The collection 𝐾, which consists of functions 𝐹(𝑧)(𝑤) = ⟨𝑤𝑇(𝑧)⟩ = ⟨𝑇(𝑤) ∣ 𝑧 for each 𝑧 in the unit ball, is a collection of bounded linear functions by ipblnfi 28287, so by the Uniform Boundedness theorem ubth 28305, 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)) ≤ 𝑦𝑇(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 = (·𝑖OLD𝑈)    &   𝐿 = (𝑈 LnOp 𝑈)    &   𝐵 = (𝑈 BLnOp 𝑈)    &   𝑁 = (normCV𝑈)    &   𝑈 ∈ CHilOLD    &   𝑊 = ⟨⟨ + , · ⟩, abs⟩    &   (𝜑𝑇𝐿)    &   (𝜑 → ∀𝑥𝑋𝑦𝑋 (𝑥𝑃(𝑇𝑦)) = ((𝑇𝑥)𝑃𝑦))    &   𝐹 = (𝑧𝑋 ↦ (𝑤𝑋 ↦ (𝑤𝑃(𝑇𝑧))))    &   𝐾 = (𝐹 “ {𝑧𝑋 ∣ (𝑁𝑧) ≤ 1})       (𝜑𝑇𝐵)

Theoremhtth 28351* 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 = (·𝑖OLD𝑈)    &   𝐿 = (𝑈 LnOp 𝑈)    &   𝐵 = (𝑈 BLnOp 𝑈)       ((𝑈 ∈ CHilOLD𝑇𝐿 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑃(𝑇𝑦)) = ((𝑇𝑥)𝑃𝑦)) → 𝑇𝐵)

PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)

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 20451; note that df-hil 20451 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.

19.1  Axiomatization of complex pre-Hilbert spaces

19.1.1  Basic Hilbert space definitions

Syntaxchba 28352 Extend class notation with Hilbert vector space.
class

Syntaxcva 28353 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 10277.
class +

Syntaxcsm 28354 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 ·

Syntaxcsp 28355 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 4405.
class ·ih

Syntaxcno 28356 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 norm

Syntaxc0v 28357 Extend class notation with zero vector in Hilbert space.
class 0

Syntaxcmv 28358 Extend class notation with vector subtraction in Hilbert space.
class

Syntaxccauold 28359 Extend class notation with set of Cauchy sequences in Hilbert space.
class Cauchy

Syntaxchli 28360 Extend class notation with convergence relation in Hilbert space.
class 𝑣

Syntaxcsh 28361 Extend class notation with set of subspaces of a Hilbert space.
class S

Syntaxcch 28362 Extend class notation with set of closed subspaces of a Hilbert space.
class C

Syntaxcort 28363 Extend class notation with orthogonal complement in C.
class

Syntaxcph 28364 Extend class notation with subspace sum in C.
class +

Syntaxcspn 28365 Extend class notation with subspace span in C.
class span

Syntaxchj 28366 Extend class notation with join in C.
class

Syntaxchsup 28367 Extend class notation with supremum of a collection in C.
class

Syntaxc0h 28368 Extend class notation with zero of C.
class 0

Syntaxccm 28369 Extend class notation with the commutes relation on a Hilbert lattice.
class 𝐶

Syntaxcpjh 28370 Extend class notation with set of projections on a Hilbert space.
class proj

Syntaxchos 28371 Extend class notation with sum of Hilbert space operators.
class +op

Syntaxchot 28372 Extend class notation with scalar product of a Hilbert space operator.
class ·op

Syntaxchod 28373 Extend class notation with difference of Hilbert space operators.
class op

Syntaxchfs 28374 Extend class notation with sum of Hilbert space functionals.
class +fn

Syntaxchft 28375 Extend class notation with scalar product of Hilbert space functional.
class ·fn

Syntaxch0o 28376 Extend class notation with the Hilbert space zero operator.
class 0hop

Syntaxchio 28377 Extend class notation with Hilbert space identity operator.
class Iop

Syntaxcnop 28378 Extend class notation with the operator norm function.
class normop

Syntaxccop 28379 Extend class notation with set of continuous Hilbert space operators.
class ContOp

Syntaxclo 28380 Extend class notation with set of linear Hilbert space operators.
class LinOp

Syntaxcbo 28381 Extend class notation with set of bounded linear operators.
class BndLinOp

Syntaxcuo 28382 Extend class notation with set of unitary Hilbert space operators.
class UniOp

Syntaxcho 28383 Extend class notation with set of Hermitian Hilbert space operators.
class HrmOp

Syntaxcnmf 28384 Extend class notation with the functional norm function.
class normfn

Syntaxcnl 28385 Extend class notation with the functional nullspace function.
class null

Syntaxccnfn 28386 Extend class notation with set of continuous Hilbert space functionals.
class ContFn

Syntaxclf 28387 Extend class notation with set of linear Hilbert space functionals.
class LinFn

Syntaxcbr 28389 Extend class notation with the bra of a vector in Dirac bra-ket notation.
class bra

Syntaxck 28390 Extend class notation with the outer product of two vectors in Dirac bra-ket notation.
class ketbra

Syntaxcleo 28391 Extend class notation with positive operator ordering.
class op

Syntaxcei 28392 Extend class notation with Hilbert space eigenvector function.
class eigvec

Syntaxcel 28393 Extend class notation with Hilbert space eigenvalue function.
class eigval

Syntaxcspc 28394 Extend class notation with the spectrum of an operator.
class Lambda

Syntaxcst 28395 Extend class notation with set of states on a Hilbert lattice.
class States

Syntaxchst 28396 Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.
class CHStates

Syntaxccv 28397 Extend class notation with the covers relation on a Hilbert lattice.
class

Syntaxcat 28398 Extend class notation with set of atoms on a Hilbert lattice.
class HAtoms

Syntaxcmd 28399 Extend class notation with the modular pair relation on a Hilbert lattice.
class 𝑀

Syntaxcdmd 28400 Extend class notation with the dual modular pair relation on a Hilbert lattice.
class 𝑀*

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43667
 Copyright terms: Public domain < Previous  Next >