P. 310 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30901-31000   *Has distinct variable group(s)

Type	Label	Description
Statement
19.6.2  Examples of complex Banach spaces
Theorem	cnbn 30901	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
19.6.3  Uniform Boundedness Theorem
Theorem	ubthlem1 30902*	Lemma for ubth 30905. 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 25383, 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 30903*	Lemma for ubth 30905. 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 30904*	Lemma for ubth 30905. Prove the reverse implication, using nmblolbi 30832. (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 30905*	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))
19.6.4  Minimizing Vector Theorem
Theorem	minvecolem1 30906*	Lemma for minveco 30916. 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 30907*	Lemma for minveco 30916. 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 30908*	Lemma for minveco 30916. 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 30909*	Lemma for minveco 30916. 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 30910*	Lemma for minveco 30916. 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 30911*	Lemma for minveco 30916. 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 30912*	Lemma for minveco 30916. 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 30913*	Lemma for minveco 30916. Discharge the assumption about the sequence 𝐹 by applying countable choice ax-cc 10504. (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 30914*	Lemma for minveco 30916. 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 30915*	Lemma for minveco 30916. 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 30916*	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)))
19.7  Complex Hilbert spaces
19.7.1  Definition and basic properties
Syntax	chlo 30917	Extend class notation with the class of all complex Hilbert spaces.
class CHilOLD
Definition	df-hlo 30918	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 30919	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 30920	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 30921	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 30922	The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
⊢ Rel CHilOLD
Theorem	hlnv 30923	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 30924	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 30925	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 30926	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 30927	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 30928	The parallelogram law satisfied 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 30929	The parallelogram law satisfied 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))))
19.7.2  Standard axioms for a complex Hilbert space
Theorem	hlex 30930	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 30931	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 30932	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 30933	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 30934	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 30935	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 30936	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 30937	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 30938	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 30939	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 30940	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 30941	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 30942	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 30943	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 30944	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 30945	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 30946	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 30947	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))
19.7.3  Examples of complex Hilbert spaces
Theorem	cnchl 30948	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
19.7.4  Hellinger-Toeplitz Theorem
Theorem	htthlem 30949*	Lemma for htth 30950. 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 30887, so by the Uniform Boundedness theorem ubth 30905, 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 30950*	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)
PART 20  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 21747; note that df-hil 21747 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.
20.1  Axiomatization of complex pre-Hilbert spaces
20.1.1  Basic Hilbert space definitions
Syntax	chba 30951	Extend class notation with Hilbert vector space.
class ℋ
Syntax	cva 30952	Extend class notation with vector addition in Hilbert space. In the literature, the subscript "h" is omitted, but we need it to avoid ambiguity with complex number addition + caddc 11187.
class +ℎ
Syntax	csm 30953	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 30954	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 to use existing theorems about operations and also eliminates ambiguity with the definition of an ordered pair df-op 4655.
class ·ih
Syntax	cno 30955	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 30956	Extend class notation with zero vector in Hilbert space.
class 0ℎ
Syntax	cmv 30957	Extend class notation with vector subtraction in Hilbert space.
class −ℎ
Syntax	ccauold 30958	Extend class notation with set of Cauchy sequences in Hilbert space.
class Cauchy
Syntax	chli 30959	Extend class notation with convergence relation in Hilbert space.
class ⇝𝑣
Syntax	csh 30960	Extend class notation with set of subspaces of a Hilbert space.
class Sℋ
Syntax	cch 30961	Extend class notation with set of closed subspaces of a Hilbert space.
class Cℋ
Syntax	cort 30962	Extend class notation with orthogonal complement in Cℋ.
class ⊥
Syntax	cph 30963	Extend class notation with subspace sum in Cℋ.
class +ℋ
Syntax	cspn 30964	Extend class notation with subspace span in Cℋ.
class span
Syntax	chj 30965	Extend class notation with join in Cℋ.
class ∨ℋ
Syntax	chsup 30966	Extend class notation with supremum of a collection in Cℋ.
class ∨ℋ
Syntax	c0h 30967	Extend class notation with zero of Cℋ.
class 0ℋ
Syntax	ccm 30968	Extend class notation with the commutes relation on a Hilbert lattice.
class 𝐶ℋ
Syntax	cpjh 30969	Extend class notation with set of projections on a Hilbert space.
class projℎ
Syntax	chos 30970	Extend class notation with sum of Hilbert space operators.
class +op
Syntax	chot 30971	Extend class notation with scalar product of a Hilbert space operator.
class ·op
Syntax	chod 30972	Extend class notation with difference of Hilbert space operators.
class −op
Syntax	chfs 30973	Extend class notation with sum of Hilbert space functionals.
class +fn
Syntax	chft 30974	Extend class notation with scalar product of Hilbert space functional.
class ·fn
Syntax	ch0o 30975	Extend class notation with the Hilbert space zero operator.
class 0hop
Syntax	chio 30976	Extend class notation with Hilbert space identity operator.
class Iop
Syntax	cnop 30977	Extend class notation with the operator norm function.
class normop
Syntax	ccop 30978	Extend class notation with set of continuous Hilbert space operators.
class ContOp
Syntax	clo 30979	Extend class notation with set of linear Hilbert space operators.
class LinOp
Syntax	cbo 30980	Extend class notation with set of bounded linear operators.
class BndLinOp
Syntax	cuo 30981	Extend class notation with set of unitary Hilbert space operators.
class UniOp
Syntax	cho 30982	Extend class notation with set of Hermitian Hilbert space operators.
class HrmOp
Syntax	cnmf 30983	Extend class notation with the functional norm function.
class normfn
Syntax	cnl 30984	Extend class notation with the functional nullspace function.
class null
Syntax	ccnfn 30985	Extend class notation with set of continuous Hilbert space functionals.
class ContFn
Syntax	clf 30986	Extend class notation with set of linear Hilbert space functionals.
class LinFn
Syntax	cado 30987	Extend class notation with Hilbert space adjoint function.
class adjℎ
Syntax	cbr 30988	Extend class notation with the bra of a vector in Dirac bra-ket notation.
class bra
Syntax	ck 30989	Extend class notation with the outer product of two vectors in Dirac bra-ket notation.
class ketbra
Syntax	cleo 30990	Extend class notation with positive operator ordering.
class ≤op
Syntax	cei 30991	Extend class notation with Hilbert space eigenvector function.
class eigvec
Syntax	cel 30992	Extend class notation with Hilbert space eigenvalue function.
class eigval
Syntax	cspc 30993	Extend class notation with the spectrum of an operator.
class Lambda
Syntax	cst 30994	Extend class notation with set of states on a Hilbert lattice.
class States
Syntax	chst 30995	Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.
class CHStates
Syntax	ccv 30996	Extend class notation with the covers relation on a Hilbert lattice.
class ⋖ℋ
Syntax	cat 30997	Extend class notation with set of atoms on a Hilbert lattice.
class HAtoms
Syntax	cmd 30998	Extend class notation with the modular pair relation on a Hilbert lattice.
class 𝑀ℋ
Syntax	cdmd 30999	Extend class notation with the dual modular pair relation on a Hilbert lattice.
class 𝑀ℋ*
20.1.2  Preliminary ZFC lemmas
Definition	df-hnorm 31000	Define the function for the norm of a vector of Hilbert space. See normval 31156 for its value and normcl 31157 for its closure. Theorems norm-i-i 31165, norm-ii-i 31169, and norm-iii-i 31171 show it has the expected properties of a norm. 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. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
⊢ normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))