HomeTheorem List (p. 315 of 512)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | nmlno0i 31401 | 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 ⇒ ⊢ (𝑇 ∈ 𝐿 → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)) | ||
| Theorem | nmlno0 31402 | 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 ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)) | ||
| Theorem | nmlnoubi 31403* | 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 = (0vec‘𝑈) & ⊢ 𝐾 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐿 ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ 𝑋 (𝑥 ≠ 𝑍 → (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐾‘𝑥)))) → (𝑁‘𝑇) ≤ 𝐴) | ||
| Theorem | nmlnogt0 31404 | 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 ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ 𝑍 ↔ 0 < (𝑁‘𝑇))) | ||
| Theorem | lnon0 31405* | 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 = (0vec‘𝑈) & ⊢ 𝑂 = (𝑈 0op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ 𝑇 ≠ 𝑂) → ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍) | ||
| Theorem | nmblolbii 31406 | 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 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐵 ⇒ ⊢ (𝐴 ∈ 𝑋 → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴))) | ||
| Theorem | nmblolbi 31407 | 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 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐵 ∧ 𝐴 ∈ 𝑋) → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴))) | ||
| Theorem | isblo3i 31408* | 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 = (normCV‘𝑈) & ⊢ 𝑁 = (normCV‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (𝑀‘𝑦)))) | ||
| Theorem | blo3i 31409* | Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = (normCV‘𝑈) & ⊢ 𝑁 = (normCV‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐿 ∧ 𝐴 ∈ ℝ ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝐴 · (𝑀‘𝑦))) → 𝑇 ∈ 𝐵) | ||
| Theorem | blometi 31410 | 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 ⇒ ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → ((𝑇‘𝑃)𝐷(𝑇‘𝑄)) ≤ ((𝑁‘𝑇) · (𝑃𝐶𝑄))) | ||
| Theorem | blocnilem 31411 | Lemma for blocni 31412 and lnocni 31413. 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 & ⊢ 𝑇 ∈ 𝐿 & ⊢ 𝑋 = (BaseSet‘𝑈) ⇒ ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ 𝐵) | ||
| Theorem | blocni 31412 | 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 & ⊢ 𝑇 ∈ 𝐿 ⇒ ⊢ (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵) | ||
| Theorem | lnocni 31413 | 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 & ⊢ 𝑇 ∈ 𝐿 & ⊢ 𝑋 = (BaseSet‘𝑈) ⇒ ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | blocn 31414 | 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) ⇒ ⊢ (𝑇 ∈ 𝐿 → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵)) | ||
| Theorem | blocn2 31415 | 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 ⇒ ⊢ (𝑇 ∈ 𝐵 → 𝑇 ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | ajfval 31416* | 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 = (·𝑖OLD‘𝑈) & ⊢ 𝑄 = (·𝑖OLD‘𝑊) & ⊢ 𝐴 = (𝑈 adj 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}) | ||
| Theorem | hmoval 31417* | 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 = (𝑈 adj 𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) | ||
| Theorem | ishmo 31418 | The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝐻 = (HmOp‘𝑈) & ⊢ 𝐴 = (𝑈 adj 𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇))) | ||
| Syntax | ccphlo 31419 | Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces). |
| class CPreHilOLD | ||
| Definition | df-ph 31420* | 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 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔( -1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))}) | ||
| Theorem | phnv 31421 | Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | ||
| Theorem | phrel 31422 | The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ Rel CPreHilOLD | ||
| Theorem | phnvi 31423 | 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 | ||
| Theorem | isphg 31424* | 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 ⇒ ⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺( -1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))) | ||
| Theorem | phop 31425 | A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩) | ||
| Theorem | cncph 31426 | 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 | ||
| Theorem | elimph 31427 | 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 = (0vec‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋 | ||
| Theorem | elimphu 31428 | 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 | ||
| Theorem | isph 31429* | The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ (𝑈 ∈ CPreHilOLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))) | ||
| Theorem | phpar2 31430 | The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | ||
| Theorem | phpar 31431 | The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺( -1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | ||
| Theorem | ip0i 31432 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 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))) | ||
| Theorem | ip1ilem 31433 | Lemma for ip1i 31434. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐽 ∈ ℂ ⇒ ⊢ (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺( -1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶)) | ||
| Theorem | ip1i 31434 | Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 ⇒ ⊢ (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺( -1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶)) | ||
| Theorem | ip2i 31435 | Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((2𝑆𝐴)𝑃𝐵) = (2 · (𝐴𝑃𝐵)) | ||
| Theorem | ipdirilem 31436 | Lemma for ipdiri 31437. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 ⇒ ⊢ ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)) | ||
| Theorem | ipdiri 31437 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶))) | ||
| Theorem | ipasslem1 31438 | Lemma for ipassi 31448. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem2 31439 | Lemma for ipassi 31448. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → (( -𝑁𝑆𝐴)𝑃𝐵) = ( -𝑁 · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem3 31440 | Lemma for ipassi 31448. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem4 31441 | Lemma for ipassi 31448. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ 𝑋) → (((1 / 𝑁)𝑆𝐴)𝑃𝐵) = ((1 / 𝑁) · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem5 31442 | Lemma for ipassi 31448. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝐶 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem7 31443* | Lemma for ipassi 31448. Show that ((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)) is continuous on ℝ. (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐹 ∈ (𝐽 Cn 𝐾) | ||
| Theorem | ipasslem8 31444* | Lemma for ipassi 31448. By ipasslem5 31442, 𝐹 is 0 for all ℚ; since it is continuous and ℚ is dense in ℝ by qdensere2 25121, we conclude 𝐹 is 0 for all ℝ. (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ⇒ ⊢ 𝐹:ℝ⟶{0} | ||
| Theorem | ipasslem9 31445 | Lemma for ipassi 31448. Conclude from ipasslem8 31444 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ (𝐶 ∈ ℝ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem10 31446 | Lemma for ipassi 31448. 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((i𝑆𝐴)𝑃𝐵) = (i · (𝐴𝑃𝐵)) | ||
| Theorem | ipasslem11 31447 | Lemma for ipassi 31448. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ (𝐶 ∈ ℂ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵))) | ||
| Theorem | ipassi 31448 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))) | ||
| Theorem | dipdir 31449 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶))) | ||
| Theorem | dipdi 31450 | Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝐺𝐶)) = ((𝐴𝑃𝐵) + (𝐴𝑃𝐶))) | ||
| Theorem | ip2dii 31451 | Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 & ⊢ 𝐷 ∈ 𝑋 ⇒ ⊢ ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = (((𝐴𝑃𝐶) + (𝐵𝑃𝐷)) + ((𝐴𝑃𝐷) + (𝐵𝑃𝐶))) | ||
| Theorem | dipass 31452 | 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 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))) | ||
| Theorem | dipassr 31453 | "Associative" law for second argument of inner product (compare dipass 31452). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑆𝐶)) = ((∗‘𝐵) · (𝐴𝑃𝐶))) | ||
| Theorem | dipassr2 31454 | "Associative" law for inner product. Conjugate version of dipassr 31453. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃((∗‘𝐵)𝑆𝐶)) = (𝐵 · (𝐴𝑃𝐶))) | ||
| Theorem | dipsubdir 31455 | Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑀𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) − (𝐵𝑃𝐶))) | ||
| Theorem | dipsubdi 31456 | Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑀𝐶)) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) | ||
| Theorem | pythi 31457 | 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. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝐴𝑃𝐵) = 0 → ((𝑁‘(𝐴𝐺𝐵))↑2) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))) | ||
| Theorem | siilem1 31458 | Lemma for sii 31461. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝐶 ∈ ℂ & ⊢ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ & ⊢ 0 ≤ (𝐶 · (𝐴𝑃𝐵)) ⇒ ⊢ ((𝐵𝑃𝐴) = (𝐶 · ((𝑁‘𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁‘𝐵)↑2)))) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵))) | ||
| Theorem | siilem2 31459 | Lemma for sii 31461. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))) → ((𝐵𝑃𝐴) = (𝐶 · ((𝑁‘𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁‘𝐵)↑2)))) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵)))) | ||
| Theorem | siii 31460 | Inference from sii 31461. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵)) | ||
| Theorem | sii 31461 | Obsolete version of ipcau 25564 as of 22-Sep-2024. 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 31727, bcsiALT 31786, bcsiHIL 31787, csbren 25725. (Contributed by NM, 12-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵))) | ||
| Theorem | ipblnfi 31462* | 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 ℂ. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐶 = ⟨⟨ + , · ⟩, abs⟩ & ⊢ 𝐵 = (𝑈 BLnOp 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑥𝑃𝐴)) ⇒ ⊢ (𝐴 ∈ 𝑋 → 𝐹 ∈ 𝐵) | ||
| Theorem | ip2eqi 31463* | 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 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑥𝑃𝐴) = (𝑥𝑃𝐵) ↔ 𝐴 = 𝐵)) | ||
| Theorem | phoeqi 31464* | A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ ((𝑆:𝑌⟶𝑋 ∧ 𝑇:𝑌⟶𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑆‘𝑦)) = (𝑥𝑃(𝑇‘𝑦)) ↔ 𝑆 = 𝑇)) | ||
| Theorem | ajmoi 31465* | Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ ∃*𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) | ||
| Theorem | ajfuni 31466 | The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝐴 = (𝑈 adj 𝑊) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ Fun 𝐴 | ||
| Theorem | ajfun 31467 | The adjoint function is a function. This is not immediately apparent from df-aj 31357 but results from the uniqueness shown by ajmoi 31465. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝐴 = (𝑈 adj 𝑊) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec) → Fun 𝐴) | ||
| Theorem | ajval 31468* | Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑄 = (·𝑖OLD‘𝑊) & ⊢ 𝐴 = (𝑈 adj 𝑊) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝐴‘𝑇) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))) | ||
| Syntax | ccbn 31469 | Extend class notation with the class of all complex Banach spaces. |
| class CBan | ||
| Definition | df-cbn 31470 | Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) Use df-bn 25662 instead. (New usage is discouraged.) |
| ⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))} | ||
| Theorem | iscbn 31471 | A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 25664 instead. (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋))) | ||
| Theorem | cbncms 31472 | The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 25673 (or preferably bncms 25670) instead. (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋)) | ||
| Theorem | bnnv 31473 | Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) Use bnnvc 25666 instead. (New usage is discouraged.) |
| ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec) | ||
| Theorem | bnrel 31474 | The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
| ⊢ Rel CBan | ||
| Theorem | bnsscmcl 31475 | 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 ∧ 𝑊 ∈ 𝐻) → (𝑊 ∈ CBan ↔ 𝑌 ∈ (Clsd‘𝐽))) | ||
| Theorem | cnbn 31476 | 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 | ||
| Theorem | ubthlem1 31477* | Lemma for ubth 31480. 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 25656, 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 31478* | Lemma for ubth 31480. 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 31479* | Lemma for ubth 31480. Prove the reverse implication, using nmblolbi 31407. (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 31480* | 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)) | ||
| Theorem | minvecolem1 31481* | Lemma for minveco 31491. 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 31482* | Lemma for minveco 31491. 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 31483* | Lemma for minveco 31491. 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 31484* | Lemma for minveco 31491. 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 31485* | Lemma for minveco 31491. 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 31486* | Lemma for minveco 31491. 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 31487* | Lemma for minveco 31491. 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 31488* | Lemma for minveco 31491. Discharge the assumption about the sequence 𝐹 by applying countable choice ax-cc 10516. (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 31489* | Lemma for minveco 31491. 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 31490* | Lemma for minveco 31491. 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 31491* | 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))) | ||
| Syntax | chlo 31492 | Extend class notation with the class of all complex Hilbert spaces. |
| class CHilOLD | ||
| Definition | df-hlo 31493 | 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 31494 | 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 31495 | 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 31496 | 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 31497 | The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
| ⊢ Rel CHilOLD | ||
| Theorem | hlnv 31498 | 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 31499 | 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 31500 | 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) | ||
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