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Theorem List for Metamath Proof Explorer - 26801-26900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembloln 26801 A bounded operator is a linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
𝐿 = (𝑈 LnOp 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐵) → 𝑇𝐿)
 
Theoremblof 26802 A bounded operator is an operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐵) → 𝑇:𝑋𝑌)
 
Theoremnmblore 26803 The norm of a bounded operator is a real number. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐵) → (𝑁𝑇) ∈ ℝ)
 
Theorem0ofval 26804 The zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑊)    &   𝑂 = (𝑈 0op 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))
 
Theorem0oval 26805 Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑊)    &   𝑂 = (𝑈 0op 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑂𝐴) = 𝑍)
 
Theorem0oo 26806 The zero operator is an operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝑍 = (𝑈 0op 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:𝑋𝑌)
 
Theorem0lno 26807 The zero operator is linear. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
𝑍 = (𝑈 0op 𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍𝐿)
 
Theoremnmoo0 26808 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 26809 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 26810 Lemma for nmlno0i 26811. (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 26811 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 26812 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 26813* 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 26814 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 26815* 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 26816 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 26817 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 26818* 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 26819* 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 26820 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 26821 Lemma for blocni 26822 and lnocni 26823. 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 26822 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 26823 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 26824 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 26825 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 26826* 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 26827* 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 26828 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 26829 Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).
class CPreHilOLD
 
Definitiondf-ph 26830* 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 26831 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 26832 The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Rel CPreHilOLD
 
Theoremphnvi 26833 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 26834* 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 26835 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 26836 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 26837 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 26838 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 26839* 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 26840 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 26841 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 26842 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 26843 Lemma for ip1i 26844. (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 26844 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 26845 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 26846 Lemma for ipdiri 26847. (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 26847 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 26848 Lemma for ipassi 26858. 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 26849 Lemma for ipassi 26858. 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 26850 Lemma for ipassi 26858. 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 26851 Lemma for ipassi 26858. 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 26852 Lemma for ipassi 26858. 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 26853* Lemma for ipassi 26858. 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 26854* Lemma for ipassi 26858. By ipasslem5 26852, 𝐹 is 0 for all ; since it is continuous and is dense in by qdensere2 22317, 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 26855 Lemma for ipassi 26858. Conclude from ipasslem8 26854 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 26856 Lemma for ipassi 26858. 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 26857 Lemma for ipassi 26858. 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 26858 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 26859 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 26860 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 26861 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 26862 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 26863 "Associative" law for second argument of inner product (compare dipass 26862). (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 26864 "Associative" law for inner product. Conjugate version of dipassr 26863. (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 26865 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 26866 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 26867 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 26868 Lemma for sii 26871. (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 26869 Lemma for sii 26871. (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 26870 Inference from sii 26871. (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 26871 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 27149, bcsiALT 27208, bcsiHIL 27209, csbren 22853. 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)))
 
Theoremsspph 26872 A subspace of an inner product space is an inner product space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
𝐻 = (SubSp‘𝑈)       ((𝑈 ∈ CPreHilOLD𝑊𝐻) → 𝑊 ∈ CPreHilOLD)
 
Theoremipblnfi 26873* 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 26874* 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 26875* 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 26876* 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 26877 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 26878 The adjoint function is a function. This is not immediately apparent from df-aj 26767 but results from the uniqueness shown by ajmoi 26876. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
𝐴 = (𝑈adj𝑊)       ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec) → Fun 𝐴)
 
Theoremajval 26879* 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 26880 Extend class notation with the class of all complex Banach spaces.
class CBan
 
Definitiondf-cbn 26881 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 26882 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 26883 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 26884 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 26885 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
Rel CBan
 
Theorembnsscmcl 26886 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 26887 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 26888* Lemma for ubth 26891. 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 22798, 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 26889* Lemma for ubth 26891. 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 26890* Lemma for ubth 26891. Prove the reverse implication, using nmblolbi 26817. (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 26891* 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 26892* Lemma for minveco 26902. 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 26893* Lemma for minveco 26902. 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 26894* Lemma for minveco 26902. 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 26895* Lemma for minveco 26902. 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 26896* Lemma for minveco 26902. 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 26897* Lemma for minveco 26902. 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 26898* Lemma for minveco 26902. 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 26899* Lemma for minveco 26902. Discharge the assumption about the sequence 𝐹 by applying countable choice ax-cc 9016. (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 26900* Lemma for minveco 26902. 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))))
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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 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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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