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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nvclvec 22501 | A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LVec) | ||
Theorem | nvclmod 22502 | A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LMod) | ||
Theorem | isnvc2 22503 | 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 22504 | A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ TopVec) | ||
Theorem | lssnlm 22505 | 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 22506 | 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 22507 | 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 22508 | 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 22509 | The operator norm function. |
class normOp | ||
Syntax | cnghm 22510 | The class of normed group homomorphisms. |
class NGHom | ||
Syntax | cnmhm 22511 | The class of normed module homomorphisms. |
class NMHom | ||
Definition | df-nmo 22512* | 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 22513* | 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 22514* | 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 22515 | 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 22516 | Lemma for normed group homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ Rel dom NGHom | ||
Theorem | reldmnmhm 22517 | Lemma for module homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ Rel dom NMHom | ||
Theorem | nmofval 22518* | 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 22519* | 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 22520* | 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 22521* | 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 22522* | 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 22523 | 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 22524 | 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 22525 | 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 22526 | 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 22527 | 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 22528 | 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 22529 | 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 22530 | 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 22531 | 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 22532 | 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 22533 | 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 22534 | 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 22535* | 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 22536 | Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp) | ||
Theorem | nghmrcl2 22537 | Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp) | ||
Theorem | nghmghm 22538 | 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 22539 | 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 22540 | 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 22541 | 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 22542 | 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 22543 | 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 22544 | 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 22545 | 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 22546 | 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 22547 | 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 22548 | 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 22549 | 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 22550 | 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 22551 | Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑆 ∈ NrmMod) | ||
Theorem | nmhmrcl2 22552 | Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑇 ∈ NrmMod) | ||
Theorem | nmhmlmhm 22553 | 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 22554 | 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 22555 | 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 22556 | 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 22557 | 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 22558 | 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 22559 | 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 22560 | 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 22561 | 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 22562 | 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 22563 | 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 22564 | 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 22565 | The standard topology on the reals. (Contributed by FL, 4-Jun-2007.) |
⊢ (topGen‘ran (,)) ∈ Top | ||
Theorem | uniretop 22566 | The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.) |
⊢ ℝ = ∪ (topGen‘ran (,)) | ||
Theorem | retopon 22567 | 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 22568 | The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.) |
⊢ 𝐾 = {〈(Base‘ndx), ℝ〉, 〈(TopSet‘ndx), (topGen‘ran (,))〉} ⇒ ⊢ 𝐾 ∈ TopSp | ||
Theorem | iooretop 22569 | Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) |
⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) | ||
Theorem | icccld 22570 | Closed intervals are closed sets of the standard topology on ℝ. (Contributed by FL, 14-Sep-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,)))) | ||
Theorem | icopnfcld 22571 | Right-unbounded closed intervals are closed sets of the standard topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.) |
⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) | ||
Theorem | iocmnfcld 22572 | Left-unbounded closed intervals are closed sets of the standard topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.) |
⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,)))) | ||
Theorem | qdensere 22573 | ℚ is dense in the standard topology on ℝ. (Contributed by NM, 1-Mar-2007.) |
⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ | ||
Theorem | cnmetdval 22574 | Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.) |
⊢ 𝐷 = (abs ∘ − ) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵))) | ||
Theorem | cnmet 22575 | The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.) |
⊢ (abs ∘ − ) ∈ (Met‘ℂ) | ||
Theorem | cnxmet 22576 | The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.) |
⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | ||
Theorem | cnbl0 22577 | Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.) |
⊢ 𝐷 = (abs ∘ − ) ⇒ ⊢ (𝑅 ∈ ℝ* → (◡abs “ (0[,)𝑅)) = (0(ball‘𝐷)𝑅)) | ||
Theorem | cnblcld 22578* | Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.) |
⊢ 𝐷 = (abs ∘ − ) ⇒ ⊢ (𝑅 ∈ ℝ* → (◡abs “ (0[,]𝑅)) = {𝑥 ∈ ℂ ∣ (0𝐷𝑥) ≤ 𝑅}) | ||
Theorem | cnfldms 22579 | The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
⊢ ℂfld ∈ MetSp | ||
Theorem | cnfldxms 22580 | The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
⊢ ℂfld ∈ ∞MetSp | ||
Theorem | cnfldtps 22581 | The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
⊢ ℂfld ∈ TopSp | ||
Theorem | cnfldnm 22582 | The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ abs = (norm‘ℂfld) | ||
Theorem | cnngp 22583 | The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ ℂfld ∈ NrmGrp | ||
Theorem | cnnrg 22584 | The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ ℂfld ∈ NrmRing | ||
Theorem | cnfldtopn 22585 | The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.) |
⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) | ||
Theorem | cnfldtopon 22586 | The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.) |
⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ (TopOn‘ℂ) | ||
Theorem | cnfldtop 22587 | The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.) |
⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ Top | ||
Theorem | cnfldhaus 22588 | The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.) |
⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ Haus | ||
Theorem | unicntop 22589 | The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ℂ = ∪ (TopOpen‘ℂfld) | ||
Theorem | cnopn 22590 | The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ℂ ∈ (TopOpen‘ℂfld) | ||
Theorem | zringnrg 22591 | The ring of integers is a normed ring. (Contributed by AV, 13-Jun-2019.) |
⊢ ℤring ∈ NrmRing | ||
Theorem | remetdval 22592 | Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.) |
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵))) | ||
Theorem | remet 22593 | The absolute value metric determines a metric space on the reals. (Contributed by NM, 10-Feb-2007.) |
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) ⇒ ⊢ 𝐷 ∈ (Met‘ℝ) | ||
Theorem | rexmet 22594 | The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.) |
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) ⇒ ⊢ 𝐷 ∈ (∞Met‘ℝ) | ||
Theorem | bl2ioo 22595 | A ball in terms of an open interval of reals. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.) |
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(ball‘𝐷)𝐵) = ((𝐴 − 𝐵)(,)(𝐴 + 𝐵))) | ||
Theorem | ioo2bl 22596 | An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.) |
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(,)𝐵) = (((𝐴 + 𝐵) / 2)(ball‘𝐷)((𝐵 − 𝐴) / 2))) | ||
Theorem | ioo2blex 22597 | An open interval of reals in terms of a ball. (Contributed by Mario Carneiro, 14-Nov-2013.) |
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(,)𝐵) ∈ ran (ball‘𝐷)) | ||
Theorem | blssioo 22598 | The balls of the standard real metric space are included in the open real intervals. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.) |
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) ⇒ ⊢ ran (ball‘𝐷) ⊆ ran (,) | ||
Theorem | tgioo 22599 | The topology generated by open intervals of reals is the same as the open sets of the standard metric space on the reals. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.) |
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ (topGen‘ran (,)) = 𝐽 | ||
Theorem | qdensere2 22600 | ℚ is dense in ℝ. (Contributed by NM, 24-Aug-2007.) |
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ((cls‘𝐽)‘ℚ) = ℝ |
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