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Theorem List for Metamath Proof Explorer - 28201-28300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnmoo0 28201 The operator norm of the zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)
𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑍 = (𝑈 0op 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁𝑍) = 0)

Theorem0blo 28202 The zero operator is a bounded linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
𝑍 = (𝑈 0op 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍𝐵)

Theoremnmlno0lem 28203 Lemma for nmlno0i 28204. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑍 = (𝑈 0op 𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec    &   𝑇𝐿    &   𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝑅 = ( ·𝑠OLD𝑈)    &   𝑆 = ( ·𝑠OLD𝑊)    &   𝑃 = (0vec𝑈)    &   𝑄 = (0vec𝑊)    &   𝐾 = (normCV𝑈)    &   𝑀 = (normCV𝑊)       ((𝑁𝑇) = 0 ↔ 𝑇 = 𝑍)

Theoremnmlno0i 28204 The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑍 = (𝑈 0op 𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       (𝑇𝐿 → ((𝑁𝑇) = 0 ↔ 𝑇 = 𝑍))

Theoremnmlno0 28205 The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑍 = (𝑈 0op 𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → ((𝑁𝑇) = 0 ↔ 𝑇 = 𝑍))

Theoremnmlnoubi 28206* An upper bound for the operator norm of a linear operator, using only the properties of nonzero arguments. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)    &   𝐾 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇𝐿 ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥𝑋 (𝑥𝑍 → (𝑀‘(𝑇𝑥)) ≤ (𝐴 · (𝐾𝑥)))) → (𝑁𝑇) ≤ 𝐴)

Theoremnmlnogt0 28207 The norm of a nonzero linear operator is positive. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)
𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑍 = (𝑈 0op 𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → (𝑇𝑍 ↔ 0 < (𝑁𝑇)))

Theoremlnon0 28208* The domain of a nonzero linear operator contains a nonzero vector. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)    &   𝑂 = (𝑈 0op 𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)       (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ 𝑇𝑂) → ∃𝑥𝑋 𝑥𝑍)

Theoremnmblolbii 28209 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐿 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec    &   𝑇𝐵       (𝐴𝑋 → (𝑀‘(𝑇𝐴)) ≤ ((𝑁𝑇) · (𝐿𝐴)))

Theoremnmblolbi 28210 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐿 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇𝐵𝐴𝑋) → (𝑀‘(𝑇𝐴)) ≤ ((𝑁𝑇) · (𝐿𝐴)))

Theoremisblo3i 28211* The predicate "is a bounded linear operator." Definition 2.7-1 of [Kreyszig] p. 91. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = (normCV𝑈)    &   𝑁 = (normCV𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       (𝑇𝐵 ↔ (𝑇𝐿 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑋 (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (𝑀𝑦))))

Theoremblo3i 28212* Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = (normCV𝑈)    &   𝑁 = (normCV𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇𝐿𝐴 ∈ ℝ ∧ ∀𝑦𝑋 (𝑁‘(𝑇𝑦)) ≤ (𝐴 · (𝑀𝑦))) → 𝑇𝐵)

Theoremblometi 28213 Upper bound for the distance between the values of a bounded linear operator. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝐶 = (IndMet‘𝑈)    &   𝐷 = (IndMet‘𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇𝐵𝑃𝑋𝑄𝑋) → ((𝑇𝑃)𝐷(𝑇𝑄)) ≤ ((𝑁𝑇) · (𝑃𝐶𝑄)))

Theoremblocnilem 28214 Lemma for blocni 28215 and lnocni 28216. If a linear operator is continuous at any point, it is bounded. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
𝐶 = (IndMet‘𝑈)    &   𝐷 = (IndMet‘𝑊)    &   𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   𝐿 = (𝑈 LnOp 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec    &   𝑇𝐿    &   𝑋 = (BaseSet‘𝑈)       ((𝑃𝑋𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇𝐵)

Theoremblocni 28215 A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
𝐶 = (IndMet‘𝑈)    &   𝐷 = (IndMet‘𝑊)    &   𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   𝐿 = (𝑈 LnOp 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec    &   𝑇𝐿       (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇𝐵)

Theoremlnocni 28216 If a linear operator is continuous at any point, it is continuous everywhere. Theorem 2.7-9(b) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (New usage is discouraged.)
𝐶 = (IndMet‘𝑈)    &   𝐷 = (IndMet‘𝑊)    &   𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   𝐿 = (𝑈 LnOp 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec    &   𝑇𝐿    &   𝑋 = (BaseSet‘𝑈)       ((𝑃𝑋𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ (𝐽 Cn 𝐾))

Theoremblocn 28217 A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.)
𝐶 = (IndMet‘𝑈)    &   𝐷 = (IndMet‘𝑊)    &   𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec    &   𝐿 = (𝑈 LnOp 𝑊)       (𝑇𝐿 → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇𝐵))

Theoremblocn2 28218 A bounded linear operator is continuous. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.)
𝐶 = (IndMet‘𝑈)    &   𝐷 = (IndMet‘𝑊)    &   𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       (𝑇𝐵𝑇 ∈ (𝐽 Cn 𝐾))

Theoremajfval 28219* The adjoint function. (Contributed by NM, 25-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑄 = (·𝑖OLD𝑊)    &   𝐴 = (𝑈adj𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))})

Theoremhmoval 28220* The set of Hermitian (self-adjoint) operators on a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
𝐻 = (HmOp‘𝑈)    &   𝐴 = (𝑈adj𝑈)       (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴𝑡) = 𝑡})

Theoremishmo 28221 The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
𝐻 = (HmOp‘𝑈)    &   𝐴 = (𝑈adj𝑈)       (𝑈 ∈ NrmCVec → (𝑇𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴𝑇) = 𝑇)))

18.5  Inner product (pre-Hilbert) spaces

18.5.1  Definition and basic properties

Syntaxccphlo 28222 Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).
class CPreHilOLD

Definitiondf-ph 28223* Define the class of all complex inner product spaces. An inner product space is a normed vector space whose norm satisfies the parallelogram law (a property that induces an inner product). Based on Exercise 4(b) of [ReedSimon] p. 63. The vector operation is 𝑔, the scalar product is 𝑠, and the norm is 𝑛. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
CPreHilOLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))})

Theoremphnv 28224 Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
(𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)

Theoremphrel 28225 The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Rel CPreHilOLD

Theoremphnvi 28226 Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
𝑈 ∈ CPreHilOLD       𝑈 ∈ NrmCVec

Theoremisphg 28227* The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is 𝐺, the scalar product is 𝑆, and the norm is 𝑁. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺𝐴𝑆𝐵𝑁𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))))

Theoremphop 28228 A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)       (𝑈 ∈ CPreHilOLD𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

18.5.2  Examples of pre-Hilbert spaces

Theoremcncph 28229 The set of complex numbers is an inner product (pre-Hilbert) space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 7-Nov-2013.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, abs⟩       𝑈 ∈ CPreHilOLD

Theoremelimph 28230 Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)    &   𝑈 ∈ CPreHilOLD       if(𝐴𝑋, 𝐴, 𝑍) ∈ 𝑋

Theoremelimphu 28231 Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 6-May-2007.) (New usage is discouraged.)
if(𝑈 ∈ CPreHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ CPreHilOLD

18.5.3  Properties of pre-Hilbert spaces

Theoremisph 28232* The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)       (𝑈 ∈ CPreHilOLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2)))))

Theoremphpar2 28233 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ CPreHilOLD𝐴𝑋𝐵𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))

Theoremphpar 28234 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ CPreHilOLD𝐴𝑋𝐵𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))

Theoremip0i 28235 A slight variant of Equation 6.46 of [Ponnusamy] p. 362, where 𝐽 is either 1 or -1 to represent +-1. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋    &   𝑁 = (normCV𝑈)    &   𝐽 ∈ ℂ       ((((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2)) + (((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2))) = (2 · (((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)))

Theoremip1ilem 28236 Lemma for ip1i 28237. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋    &   𝑁 = (normCV𝑈)    &   𝐽 ∈ ℂ       (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺(-1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶))

Theoremip1i 28237 Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋       (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺(-1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶))

Theoremip2i 28238 Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       ((2𝑆𝐴)𝑃𝐵) = (2 · (𝐴𝑃𝐵))

Theoremipdirilem 28239 Lemma for ipdiri 28240. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋       ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶))

Theoremipdiri 28240 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝐴𝑋𝐵𝑋𝐶𝑋) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Theoremipasslem1 28241 Lemma for ipassi 28251. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝑁 ∈ ℕ0𝐴𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Theoremipasslem2 28242 Lemma for ipassi 28251. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝑁 ∈ ℕ0𝐴𝑋) → ((-𝑁𝑆𝐴)𝑃𝐵) = (-𝑁 · (𝐴𝑃𝐵)))

Theoremipasslem3 28243 Lemma for ipassi 28251. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝑁 ∈ ℤ ∧ 𝐴𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Theoremipasslem4 28244 Lemma for ipassi 28251. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝑁 ∈ ℕ ∧ 𝐴𝑋) → (((1 / 𝑁)𝑆𝐴)𝑃𝐵) = ((1 / 𝑁) · (𝐴𝑃𝐵)))

Theoremipasslem5 28245 Lemma for ipassi 28251. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝐶 ∈ ℚ ∧ 𝐴𝑋) → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))

Theoremipasslem7 28246* Lemma for ipassi 28251. Show that ((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)) is continuous on . (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))))    &   𝐽 = (topGen‘ran (,))    &   𝐾 = (TopOpen‘ℂfld)       𝐹 ∈ (𝐽 Cn 𝐾)

Theoremipasslem8 28247* Lemma for ipassi 28251. By ipasslem5 28245, 𝐹 is 0 for all ; since it is continuous and is dense in by qdensere2 22970, we conclude 𝐹 is 0 for all . (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))))       𝐹:ℝ⟶{0}

Theoremipasslem9 28248 Lemma for ipassi 28251. Conclude from ipasslem8 28247 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       (𝐶 ∈ ℝ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))

Theoremipasslem10 28249 Lemma for ipassi 28251. Show the inner product associative law for the imaginary number i. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝑁 = (normCV𝑈)       ((i𝑆𝐴)𝑃𝐵) = (i · (𝐴𝑃𝐵))

Theoremipasslem11 28250 Lemma for ipassi 28251. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       (𝐶 ∈ ℂ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))

Theoremipassi 28251 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Theoremdipdir 28252 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Theoremdipdi 28253 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑃(𝐵𝐺𝐶)) = ((𝐴𝑃𝐵) + (𝐴𝑃𝐶)))

Theoremip2dii 28254 Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋    &   𝐷𝑋       ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = (((𝐴𝑃𝐶) + (𝐵𝑃𝐷)) + ((𝐴𝑃𝐷) + (𝐵𝑃𝐶)))

Theoremdipass 28255 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Theoremdipassr 28256 "Associative" law for second argument of inner product (compare dipass 28255). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵 ∈ ℂ ∧ 𝐶𝑋)) → (𝐴𝑃(𝐵𝑆𝐶)) = ((∗‘𝐵) · (𝐴𝑃𝐶)))

Theoremdipassr2 28257 "Associative" law for inner product. Conjugate version of dipassr 28256. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵 ∈ ℂ ∧ 𝐶𝑋)) → (𝐴𝑃((∗‘𝐵)𝑆𝐶)) = (𝐵 · (𝐴𝑃𝐶)))

Theoremdipsubdir 28258 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑀𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) − (𝐵𝑃𝐶)))

Theoremdipsubdi 28259 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑃(𝐵𝑀𝐶)) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶)))

Theorempythi 28260 The Pythagorean theorem for an arbitrary complex inner product (pre-Hilbert) space 𝑈. The square of the norm of the sum of two orthogonal vectors (i.e. whose inner product is 0) is the sum of the squares of their norms. Problem 2 in [Kreyszig] p. 135. This is Metamath 100 proof #4. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       ((𝐴𝑃𝐵) = 0 → ((𝑁‘(𝐴𝐺𝐵))↑2) = (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2)))

Theoremsiilem1 28261 Lemma for sii 28264. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝑀 = ( −𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝐶 ∈ ℂ    &   (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ    &   0 ≤ (𝐶 · (𝐴𝑃𝐵))       ((𝐵𝑃𝐴) = (𝐶 · ((𝑁𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁𝐵)↑2)))) ≤ ((𝑁𝐴) · (𝑁𝐵)))

Theoremsiilem2 28262 Lemma for sii 28264. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝑀 = ( −𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))) → ((𝐵𝑃𝐴) = (𝐶 · ((𝑁𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁𝐵)↑2)))) ≤ ((𝑁𝐴) · (𝑁𝐵))))

Theoremsiii 28263 Inference from sii 28264. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁𝐴) · (𝑁𝐵))

Theoremsii 28264 Schwarz inequality. Part of Lemma 3-2.1(a) of [Kreyszig] p. 137. This is also called the Cauchy-Schwarz inequality by some authors and Bunjakovaskij-Cauchy-Schwarz inequality by others. See also theorems bcseqi 28532, bcsiALT 28591, bcsiHIL 28592, csbren 23567. This is Metamath 100 proof #78. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝐴𝑋𝐵𝑋) → (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁𝐴) · (𝑁𝐵)))

TheoremsspphOLD 28265 Obsolete version of cphsscph 23419 as of 6-Oct-2022: A subspace of an inner product space is an inner product space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐻 = (SubSp‘𝑈)       ((𝑈 ∈ CPreHilOLD𝑊𝐻) → 𝑊 ∈ CPreHilOLD)

Theoremipblnfi 28266* A function 𝐹 generated by varying the first argument of an inner product (with its second argument a fixed vector 𝐴) is a bounded linear functional, i.e. a bounded linear operator from the vector space to . (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐶 = ⟨⟨ + , · ⟩, abs⟩    &   𝐵 = (𝑈 BLnOp 𝐶)    &   𝐹 = (𝑥𝑋 ↦ (𝑥𝑃𝐴))       (𝐴𝑋𝐹𝐵)

Theoremip2eqi 28267* Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋 (𝑥𝑃𝐴) = (𝑥𝑃𝐵) ↔ 𝐴 = 𝐵))

Theoremphoeqi 28268* A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝑆:𝑌𝑋𝑇:𝑌𝑋) → (∀𝑥𝑋𝑦𝑌 (𝑥𝑃(𝑆𝑦)) = (𝑥𝑃(𝑇𝑦)) ↔ 𝑆 = 𝑇))

Theoremajmoi 28269* Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ∃*𝑠(𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))

Theoremajfuni 28270 The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
𝐴 = (𝑈adj𝑊)    &   𝑈 ∈ CPreHilOLD    &   𝑊 ∈ NrmCVec       Fun 𝐴

Theoremajfun 28271 The adjoint function is a function. This is not immediately apparent from df-aj 28160 but results from the uniqueness shown by ajmoi 28269. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
𝐴 = (𝑈adj𝑊)       ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec) → Fun 𝐴)

Theoremajval 28272* 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 28273 Extend class notation with the class of all complex Banach spaces.
class CBan

Definitiondf-cbn 28274 Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) Use df-bn 23504 instead. (New usage is discouraged.)
CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}

Theoremiscbn 28275 A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 23506 instead. (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)       (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))

Theoremcbncms 28276 The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 23515 (or preferably bncms 23512) instead. (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)       (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))

Theorembnnv 28277 Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) Use bnnvc 23508 instead. (New usage is discouraged.)
(𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)

Theorembnrel 28278 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
Rel CBan

Theorembnsscmcl 28279 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 28280 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 28281* Lemma for ubth 28284. 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 23498, 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 28282* Lemma for ubth 28284. 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 28283* Lemma for ubth 28284. Prove the reverse implication, using nmblolbi 28210. (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 28284* 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 28285* Lemma for minveco 28295. 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 28286* Lemma for minveco 28295. 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 28287* Lemma for minveco 28295. 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 28288* Lemma for minveco 28295. 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 28289* Lemma for minveco 28295. 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 28290* Lemma for minveco 28295. 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 28291* Lemma for minveco 28295. 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 28292* Lemma for minveco 28295. Discharge the assumption about the sequence 𝐹 by applying countable choice ax-cc 9572. (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 28293* Lemma for minveco 28295. 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 28294* Lemma for minveco 28295. 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 28295* 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 28296 Extend class notation with the class of all complex Hilbert spaces.
class CHilOLD

Definitiondf-hlo 28297 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 28298 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 28299 Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
(𝑈 ∈ CHilOLD𝑈 ∈ CBan)

Theoremhlph 28300 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)

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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 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