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Theorem List for Metamath Proof Explorer - 27701-27800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremajval 27701* Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑄 = (·𝑖OLD𝑊)    &   𝐴 = (𝑈adj𝑊)       ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝐴𝑇) = (℩𝑠(𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))

18.6  Complex Banach spaces

18.6.1  Definition and basic properties

Syntaxccbn 27702 Extend class notation with the class of all complex Banach spaces.
class CBan

Definitiondf-cbn 27703 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))}

Theoremiscbn 27704 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)))

Theoremcbncms 27705 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))

Theorembnnv 27706 Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
(𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)

Theorembnrel 27707 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
Rel CBan

Theorembnsscmcl 27708 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 ∧ 𝑊𝐻) → (𝑊 ∈ CBan ↔ 𝑌 ∈ (Clsd‘𝐽)))

18.6.2  Examples of complex Banach spaces

Theoremcnbn 27709 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 27710* Lemma for ubth 27713. 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 23121, 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 27711* Lemma for ubth 27713. 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 27712* Lemma for ubth 27713. Prove the reverse implication, using nmblolbi 27639. (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 27713* 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 27714* Lemma for minveco 27724. 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 27715* Lemma for minveco 27724. 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 27716* Lemma for minveco 27724. 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 27717* Lemma for minveco 27724. 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 27718* Lemma for minveco 27724. 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 27719* Lemma for minveco 27724. 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 27720* Lemma for minveco 27724. 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 27721* Lemma for minveco 27724. Discharge the assumption about the sequence 𝐹 by applying countable choice ax-cc 9254. (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 27722* Lemma for minveco 27724. 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 27723* Lemma for minveco 27724. 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 27724* 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 27725 Extend class notation with the class of all complex Hilbert spaces.
class CHilOLD

Definitiondf-hlo 27726 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 27727 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 27728 Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
(𝑈 ∈ CHilOLD𝑈 ∈ CBan)

Theoremhlph 27729 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 27730 The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
Rel CHilOLD

Theoremhlnv 27731 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
(𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)

Theoremhlnvi 27732 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 27733 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 27734 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 27735 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 27736 The parallelogram law satified 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 27737 The parallelogram law satified 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 27738 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 27739 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 27740 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 27741 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 27742 The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)       (𝑈 ∈ CHilOLD𝑍𝑋)

Theoremhladdid 27743 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 27744 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 27745 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 27746 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 27747 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 27748 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 27749 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 27750 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 27751 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 27752 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 27753 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 27754 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 27755 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 27756 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

Theoremssphl 27757 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 ∈ CPreHilOLD𝑊𝐻𝑊 ∈ CBan) → 𝑊 ∈ CHilOLD)

18.7.5  Hellinger-Toeplitz Theorem

Theoremhtthlem 27758* Lemma for htth 27759. 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 27695, so by the Uniform Boundedness theorem ubth 27713, 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 27759* 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 20042; note that df-hil 20042 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

Syntaxchil 27760 Extend class notation with Hilbert vector space.
class

Syntaxcva 27761 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 9936.
class +

Syntaxcsm 27762 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 27763 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 4182.
class ·ih

Syntaxcno 27764 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 27765 Extend class notation with zero vector in Hilbert space.
class 0

Syntaxcmv 27766 Extend class notation with vector subtraction in Hilbert space.
class

Syntaxccau 27767 Extend class notation with set of Cauchy sequences in Hilbert space.
class Cauchy

Syntaxchli 27768 Extend class notation with convergence relation in Hilbert space.
class 𝑣

Syntaxcsh 27769 Extend class notation with set of subspaces of a Hilbert space.
class S

Syntaxcch 27770 Extend class notation with set of closed subspaces of a Hilbert space.
class C

Syntaxcort 27771 Extend class notation with orthogonal complement in C.
class

Syntaxcph 27772 Extend class notation with subspace sum in C.
class +

Syntaxcspn 27773 Extend class notation with subspace span in C.
class span

Syntaxchj 27774 Extend class notation with join in C.
class

Syntaxchsup 27775 Extend class notation with supremum of a collection in C.
class

Syntaxc0h 27776 Extend class notation with zero of C.
class 0

Syntaxccm 27777 Extend class notation with the commutes relation on a Hilbert lattice.
class 𝐶

Syntaxcpjh 27778 Extend class notation with set of projections on a Hilbert space.
class proj

Syntaxchos 27779 Extend class notation with sum of Hilbert space operators.
class +op

Syntaxchot 27780 Extend class notation with scalar product of a Hilbert space operator.
class ·op

Syntaxchod 27781 Extend class notation with difference of Hilbert space operators.
class op

Syntaxchfs 27782 Extend class notation with sum of Hilbert space functionals.
class +fn

Syntaxchft 27783 Extend class notation with scalar product of Hilbert space functional.
class ·fn

Syntaxch0o 27784 Extend class notation with the Hilbert space zero operator.
class 0hop

Syntaxchio 27785 Extend class notation with Hilbert space identity operator.
class Iop

Syntaxcnop 27786 Extend class notation with the operator norm function.
class normop

Syntaxccop 27787 Extend class notation with set of continuous Hilbert space operators.
class ContOp

Syntaxclo 27788 Extend class notation with set of linear Hilbert space operators.
class LinOp

Syntaxcbo 27789 Extend class notation with set of bounded linear operators.
class BndLinOp

Syntaxcuo 27790 Extend class notation with set of unitary Hilbert space operators.
class UniOp

Syntaxcho 27791 Extend class notation with set of Hermitian Hilbert space operators.
class HrmOp

Syntaxcnmf 27792 Extend class notation with the functional norm function.
class normfn

Syntaxcnl 27793 Extend class notation with the functional nullspace function.
class null

Syntaxccnfn 27794 Extend class notation with set of continuous Hilbert space functionals.
class ContFn

Syntaxclf 27795 Extend class notation with set of linear Hilbert space functionals.
class LinFn

Syntaxcbr 27797 Extend class notation with the bra of a vector in Dirac bra-ket notation.
class bra

Syntaxck 27798 Extend class notation with the outer product of two vectors in Dirac bra-ket notation.
class ketbra

Syntaxcleo 27799 Extend class notation with positive operator ordering.
class op

Syntaxcei 27800 Extend class notation with Hilbert space eigenvector function.
class eigvec

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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 42301-42322
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