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Type | Label | Description |
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Statement | ||
Theorem | nrgdsdi 24701 | Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑁 = (norm‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐷 = (dist‘𝑅) ⇒ ⊢ ((𝑅 ∈ NrmRing ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝑁‘𝐴) · (𝐵𝐷𝐶)) = ((𝐴 · 𝐵)𝐷(𝐴 · 𝐶))) | ||
Theorem | nrgdsdir 24702 | Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑁 = (norm‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐷 = (dist‘𝑅) ⇒ ⊢ ((𝑅 ∈ NrmRing ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵) · (𝑁‘𝐶)) = ((𝐴 · 𝐶)𝐷(𝐵 · 𝐶))) | ||
Theorem | nm1 24703 | The norm of one in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝑁 = (norm‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) → (𝑁‘ 1 ) = 1) | ||
Theorem | unitnmn0 24704 | The norm of a unit is nonzero in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝑁 = (norm‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝑁‘𝐴) ≠ 0) | ||
Theorem | nminvr 24705 | The norm of an inverse in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝑁 = (norm‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝑁‘(𝐼‘𝐴)) = (1 / (𝑁‘𝐴))) | ||
Theorem | nmdvr 24706 | The norm of a division in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑁 = (norm‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ / = (/r‘𝑅) ⇒ ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴 / 𝐵)) = ((𝑁‘𝐴) / (𝑁‘𝐵))) | ||
Theorem | nrgdomn 24707 | A nonzero normed ring is a domain. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝑅 ∈ NrmRing → (𝑅 ∈ Domn ↔ 𝑅 ∈ NzRing)) | ||
Theorem | nrgtgp 24708 | A normed ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp) | ||
Theorem | subrgnrg 24709 | A normed ring restricted to a subring is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) ⇒ ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐻 ∈ NrmRing) | ||
Theorem | tngnrg 24710 | Given any absolute value over a ring, augmenting the ring with the absolute value produces a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑇 = (𝑅 toNrmGrp 𝐹) & ⊢ 𝐴 = (AbsVal‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐴 → 𝑇 ∈ NrmRing) | ||
Theorem | isnlm 24711* | A normed (left) module is a module which is also a normed group over a normed ring, such that the norm distributes over scalar multiplication. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝐴 = (norm‘𝐹) ⇒ ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦)))) | ||
Theorem | nmvs 24712 | Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝐴 = (norm‘𝐹) ⇒ ⊢ ((𝑊 ∈ NrmMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌))) | ||
Theorem | nlmngp 24713 | A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | ||
Theorem | nlmlmod 24714 | A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) | ||
Theorem | nlmnrg 24715 | The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing) | ||
Theorem | nlmngp2 24716 | The scalar component of a left module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp) | ||
Theorem | nlmdsdi 24717 | Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝐷 = (dist‘𝑊) & ⊢ 𝐴 = (norm‘𝐹) ⇒ ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝐴‘𝑋) · (𝑌𝐷𝑍)) = ((𝑋 · 𝑌)𝐷(𝑋 · 𝑍))) | ||
Theorem | nlmdsdir 24718 | Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝐷 = (dist‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ 𝐸 = (dist‘𝐹) ⇒ ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → ((𝑋𝐸𝑌) · (𝑁‘𝑍)) = ((𝑋 · 𝑍)𝐷(𝑌 · 𝑍))) | ||
Theorem | nlmmul0or 24719 | If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (Revised by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝑂 = (0g‘𝐹) ⇒ ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 𝑂 ∨ 𝐵 = 0 ))) | ||
Theorem | sranlm 24720 | The subring algebra over a normed ring is a normed left module. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆) ⇒ ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmMod) | ||
Theorem | nlmvscnlem2 24721 | Lemma for nlmvscn 24723. Compare this proof with the similar elementary proof mulcn2 15628 for continuity of multiplication on ℂ. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝐷 = (dist‘𝑊) & ⊢ 𝐸 = (dist‘𝐹) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ 𝐴 = (norm‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑇 = ((𝑅 / 2) / ((𝐴‘𝐵) + 1)) & ⊢ 𝑈 = ((𝑅 / 2) / ((𝑁‘𝑋) + 𝑇)) & ⊢ (𝜑 → 𝑊 ∈ NrmMod) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝐵𝐸𝐶) < 𝑈) & ⊢ (𝜑 → (𝑋𝐷𝑌) < 𝑇) ⇒ ⊢ (𝜑 → ((𝐵 · 𝑋)𝐷(𝐶 · 𝑌)) < 𝑅) | ||
Theorem | nlmvscnlem1 24722* | Lemma for nlmvscn 24723. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝐷 = (dist‘𝑊) & ⊢ 𝐸 = (dist‘𝐹) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ 𝐴 = (norm‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑇 = ((𝑅 / 2) / ((𝐴‘𝐵) + 1)) & ⊢ 𝑈 = ((𝑅 / 2) / ((𝑁‘𝑋) + 𝑇)) & ⊢ (𝜑 → 𝑊 ∈ NrmMod) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ ℝ+ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (((𝐵𝐸𝑥) < 𝑟 ∧ (𝑋𝐷𝑦) < 𝑟) → ((𝐵 · 𝑋)𝐷(𝑥 · 𝑦)) < 𝑅)) | ||
Theorem | nlmvscn 24723 | The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 24726 and nlmtlm 24730. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·sf ‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝐾 = (TopOpen‘𝐹) ⇒ ⊢ (𝑊 ∈ NrmMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) | ||
Theorem | rlmnlm 24724 | The ring module over a normed ring is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod) | ||
Theorem | rlmnm 24725 | The norm function in the ring module. (Contributed by AV, 9-Oct-2021.) |
⊢ (norm‘𝑅) = (norm‘(ringLMod‘𝑅)) | ||
Theorem | nrgtrg 24726 | A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.) (Proof shortened by AV, 31-Oct-2024.) |
⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing) | ||
Theorem | nrginvrcnlem 24727* | Lemma for nrginvrcn 24728. Compare this proof with reccn2 15629, the elementary proof of continuity of division. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ 𝑁 = (norm‘𝑅) & ⊢ 𝐷 = (dist‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ NrmRing) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ 𝑇 = (if(1 ≤ ((𝑁‘𝐴) · 𝐵), 1, ((𝑁‘𝐴) · 𝐵)) · ((𝑁‘𝐴) / 2)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝑈 ((𝐴𝐷𝑦) < 𝑥 → ((𝐼‘𝐴)𝐷(𝐼‘𝑦)) < 𝐵)) | ||
Theorem | nrginvrcn 24728 | The ring inverse function is continuous in a normed ring. (Note that this is true even in rings which are not division rings.) (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ 𝑋 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑅) ⇒ ⊢ (𝑅 ∈ NrmRing → 𝐼 ∈ ((𝐽 ↾t 𝑈) Cn (𝐽 ↾t 𝑈))) | ||
Theorem | nrgtdrg 24729 | A normed division ring is a topological division ring. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ TopDRing) | ||
Theorem | nlmtlm 24730 | A normed module is a topological module. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ TopMod) | ||
Theorem | isnvc 24731 | A normed vector space is just a normed module which is algebraically a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) | ||
Theorem | nvcnlm 24732 | A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) | ||
Theorem | nvclvec 24733 | A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LVec) | ||
Theorem | nvclmod 24734 | A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LMod) | ||
Theorem | isnvc2 24735 | A normed vector space is just a normed module whose scalar ring is a division ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing)) | ||
Theorem | nvctvc 24736 | A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ TopVec) | ||
Theorem | lssnlm 24737 | A subspace of a normed module is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ NrmMod ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmMod) | ||
Theorem | lssnvc 24738 | A subspace of a normed vector space is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ NrmVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmVec) | ||
Theorem | rlmnvc 24739 | The ring module over a normed division ring is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → (ringLMod‘𝑅) ∈ NrmVec) | ||
Theorem | ngpocelbl 24740 | Membership of an off-center vector in a ball in a normed module. (Contributed by NM, 27-Dec-2007.) (Revised by AV, 14-Oct-2021.) |
⊢ 𝑁 = (norm‘𝐺) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋)) ⇒ ⊢ ((𝐺 ∈ NrmMod ∧ 𝑅 ∈ ℝ* ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑃 + 𝐴) ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑁‘𝐴) < 𝑅)) | ||
Syntax | cnmo 24741 | The operator norm function. |
class normOp | ||
Syntax | cnghm 24742 | The class of normed group homomorphisms. |
class NGHom | ||
Syntax | cnmhm 24743 | The class of normed module homomorphisms. |
class NMHom | ||
Definition | df-nmo 24744* | Define the norm of an operator between two normed groups (usually vector spaces). This definition produces an operator norm function for each pair of groups 〈𝑠, 𝑡〉. Equivalent to the definition of linear operator norm in [AkhiezerGlazman] p. 39. (Contributed by Mario Carneiro, 18-Oct-2015.) (Revised by AV, 25-Sep-2020.) |
⊢ normOp = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (𝑓 ∈ (𝑠 GrpHom 𝑡) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < ))) | ||
Definition | df-nghm 24745* | Define the set of normed group homomorphisms between two normed groups. A normed group homomorphism is a group homomorphism which additionally bounds the increase of norm by a fixed real operator. In vector spaces these are also known as bounded linear operators. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (◡(𝑠 normOp 𝑡) “ ℝ)) | ||
Definition | df-nmhm 24746* | Define a normed module homomorphism, also known as a bounded linear operator. This is a module homomorphism (a linear function) such that the operator norm is finite, or equivalently there is a constant 𝑐 such that... (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡))) | ||
Theorem | nmoffn 24747 | The function producing operator norm functions is a function on normed groups. (Contributed by Mario Carneiro, 18-Oct-2015.) (Proof shortened by AV, 26-Sep-2020.) |
⊢ normOp Fn (NrmGrp × NrmGrp) | ||
Theorem | reldmnghm 24748 | Lemma for normed group homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ Rel dom NGHom | ||
Theorem | reldmnmhm 24749 | Lemma for module homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ Rel dom NMHom | ||
Theorem | nmofval 24750* | Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) (Revised by AV, 26-Sep-2020.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 𝐿 = (norm‘𝑆) & ⊢ 𝑀 = (norm‘𝑇) ⇒ ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < ))) | ||
Theorem | nmoval 24751* | Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) (Revised by AV, 26-Sep-2020.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 𝐿 = (norm‘𝑆) & ⊢ 𝑀 = (norm‘𝑇) ⇒ ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘𝐹) = inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )) | ||
Theorem | nmogelb 24752* | Property of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) (Proof shortened by AV, 26-Sep-2020.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 𝐿 = (norm‘𝑆) & ⊢ 𝑀 = (norm‘𝑇) ⇒ ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ (𝑁‘𝐹) ↔ ∀𝑟 ∈ (0[,)+∞)(∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥)) → 𝐴 ≤ 𝑟))) | ||
Theorem | nmolb 24753* | Any upper bound on the values of a linear operator translates to an upper bound on the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) (Proof shortened by AV, 26-Sep-2020.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 𝐿 = (norm‘𝑆) & ⊢ 𝑀 = (norm‘𝑇) ⇒ ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) → (𝑁‘𝐹) ≤ 𝐴)) | ||
Theorem | nmolb2d 24754* | Any upper bound on the values of a linear operator at nonzero vectors translates to an upper bound on the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 𝐿 = (norm‘𝑆) & ⊢ 𝑀 = (norm‘𝑇) & ⊢ 0 = (0g‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ NrmGrp) & ⊢ (𝜑 → 𝑇 ∈ NrmGrp) & ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) ⇒ ⊢ (𝜑 → (𝑁‘𝐹) ≤ 𝐴) | ||
Theorem | nmof 24755 | The operator norm is a function into the extended reals. (Contributed by Mario Carneiro, 18-Oct-2015.) (Proof shortened by AV, 26-Sep-2020.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) ⇒ ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁:(𝑆 GrpHom 𝑇)⟶ℝ*) | ||
Theorem | nmocl 24756 | The operator norm of an operator is an extended real. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) ⇒ ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘𝐹) ∈ ℝ*) | ||
Theorem | nmoge0 24757 | The operator norm of an operator is nonnegative. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) ⇒ ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐹)) | ||
Theorem | nghmfval 24758 | A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) ⇒ ⊢ (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ) | ||
Theorem | isnghm 24759 | A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑁‘𝐹) ∈ ℝ))) | ||
Theorem | isnghm2 24760 | A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) ⇒ ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ (𝑁‘𝐹) ∈ ℝ)) | ||
Theorem | isnghm3 24761 | A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) ⇒ ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ (𝑁‘𝐹) < +∞)) | ||
Theorem | bddnghm 24762 | A bounded group homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) ⇒ ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝐴 ∈ ℝ ∧ (𝑁‘𝐹) ≤ 𝐴)) → 𝐹 ∈ (𝑆 NGHom 𝑇)) | ||
Theorem | nghmcl 24763 | A normed group homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝑁‘𝐹) ∈ ℝ) | ||
Theorem | nmoi 24764 | The operator norm achieves the minimum of the set of upper bounds, if the operator is bounded. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 𝐿 = (norm‘𝑆) & ⊢ 𝑀 = (norm‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) · (𝐿‘𝑋))) | ||
Theorem | nmoix 24765 | The operator norm is a bound on the size of an operator, even when it is infinite (using extended real multiplication). (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 𝐿 = (norm‘𝑆) & ⊢ 𝑀 = (norm‘𝑇) ⇒ ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) ·e (𝐿‘𝑋))) | ||
Theorem | nmoi2 24766 | The operator norm is a bound on the growth of a vector under the action of the operator. (Contributed by Mario Carneiro, 19-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 𝐿 = (norm‘𝑆) & ⊢ 𝑀 = (norm‘𝑇) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((𝑀‘(𝐹‘𝑋)) / (𝐿‘𝑋)) ≤ (𝑁‘𝐹)) | ||
Theorem | nmoleub 24767* | The operator norm, defined as an infimum of upper bounds, can also be defined as a supremum of norms of 𝐹(𝑥) away from zero. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 𝐿 = (norm‘𝑆) & ⊢ 𝑀 = (norm‘𝑇) & ⊢ 0 = (0g‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ NrmGrp) & ⊢ (𝜑 → 𝑇 ∈ NrmGrp) & ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴))) | ||
Theorem | nghmrcl1 24768 | Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp) | ||
Theorem | nghmrcl2 24769 | Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp) | ||
Theorem | nghmghm 24770 | A normed group homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | ||
Theorem | nmo0 24771 | The operator norm of the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑇) ⇒ ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁‘(𝑉 × { 0 })) = 0) | ||
Theorem | nmoeq0 24772 | The operator norm is zero only for the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑇) ⇒ ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → ((𝑁‘𝐹) = 0 ↔ 𝐹 = (𝑉 × { 0 }))) | ||
Theorem | nmoco 24773 | An upper bound on the operator norm of a composition. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑈) & ⊢ 𝐿 = (𝑇 normOp 𝑈) & ⊢ 𝑀 = (𝑆 normOp 𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘ 𝐺)) ≤ ((𝐿‘𝐹) · (𝑀‘𝐺))) | ||
Theorem | nghmco 24774 | The composition of normed group homomorphisms is a normed group homomorphism. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 NGHom 𝑈)) | ||
Theorem | nmotri 24775 | Triangle inequality for the operator norm. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ + = (+g‘𝑇) ⇒ ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘f + 𝐺)) ≤ ((𝑁‘𝐹) + (𝑁‘𝐺))) | ||
Theorem | nghmplusg 24776 | The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ + = (+g‘𝑇) ⇒ ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘f + 𝐺) ∈ (𝑆 NGHom 𝑇)) | ||
Theorem | 0nghm 24777 | The zero operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑇) ⇒ ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇)) | ||
Theorem | nmoid 24778 | The operator norm of the identity function on a nontrivial group. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑆) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ ((𝑆 ∈ NrmGrp ∧ { 0 } ⊊ 𝑉) → (𝑁‘( I ↾ 𝑉)) = 1) | ||
Theorem | idnghm 24779 | The identity operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑆) ⇒ ⊢ (𝑆 ∈ NrmGrp → ( I ↾ 𝑉) ∈ (𝑆 NGHom 𝑆)) | ||
Theorem | nmods 24780 | Upper bound for the distance between the values of a bounded linear operator. (Contributed by Mario Carneiro, 22-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 𝐶 = (dist‘𝑆) & ⊢ 𝐷 = (dist‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐹‘𝐴)𝐷(𝐹‘𝐵)) ≤ ((𝑁‘𝐹) · (𝐴𝐶𝐵))) | ||
Theorem | nghmcn 24781 | A normed group homomorphism is a continuous function. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ 𝐽 = (TopOpen‘𝑆) & ⊢ 𝐾 = (TopOpen‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝐽 Cn 𝐾)) | ||
Theorem | isnmhm 24782 | A normed module homomorphism is a left module homomorphism which is also a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇)))) | ||
Theorem | nmhmrcl1 24783 | Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑆 ∈ NrmMod) | ||
Theorem | nmhmrcl2 24784 | Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑇 ∈ NrmMod) | ||
Theorem | nmhmlmhm 24785 | A normed module homomorphism is a left module homomorphism (a linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇)) | ||
Theorem | nmhmnghm 24786 | A normed module homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 NGHom 𝑇)) | ||
Theorem | nmhmghm 24787 | A normed module homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | ||
Theorem | isnmhm2 24788 | A normed module homomorphism is a left module homomorphism with bounded norm (a bounded linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) ⇒ ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝑁‘𝐹) ∈ ℝ)) | ||
Theorem | nmhmcl 24789 | A normed module homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑁‘𝐹) ∈ ℝ) | ||
Theorem | idnmhm 24790 | The identity operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑆) ⇒ ⊢ (𝑆 ∈ NrmMod → ( I ↾ 𝑉) ∈ (𝑆 NMHom 𝑆)) | ||
Theorem | 0nmhm 24791 | The zero operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑇) & ⊢ 𝐹 = (Scalar‘𝑆) & ⊢ 𝐺 = (Scalar‘𝑇) ⇒ ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇)) | ||
Theorem | nmhmco 24792 | The composition of bounded linear operators is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ ((𝐹 ∈ (𝑇 NMHom 𝑈) ∧ 𝐺 ∈ (𝑆 NMHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 NMHom 𝑈)) | ||
Theorem | nmhmplusg 24793 | The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.) |
⊢ + = (+g‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 NMHom 𝑇) ∧ 𝐺 ∈ (𝑆 NMHom 𝑇)) → (𝐹 ∘f + 𝐺) ∈ (𝑆 NMHom 𝑇)) | ||
Theorem | qtopbaslem 24794 | The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.) |
⊢ 𝑆 ⊆ ℝ* ⇒ ⊢ ((,) “ (𝑆 × 𝑆)) ∈ TopBases | ||
Theorem | qtopbas 24795 | The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.) |
⊢ ((,) “ (ℚ × ℚ)) ∈ TopBases | ||
Theorem | retopbas 24796 | A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.) |
⊢ ran (,) ∈ TopBases | ||
Theorem | retop 24797 | The standard topology on the reals. (Contributed by FL, 4-Jun-2007.) |
⊢ (topGen‘ran (,)) ∈ Top | ||
Theorem | uniretop 24798 | The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.) |
⊢ ℝ = ∪ (topGen‘ran (,)) | ||
Theorem | retopon 24799 | The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.) |
⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | ||
Theorem | retps 24800 | The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.) |
⊢ 𝐾 = {〈(Base‘ndx), ℝ〉, 〈(TopSet‘ndx), (topGen‘ran (,))〉} ⇒ ⊢ 𝐾 ∈ TopSp |
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