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Theorem List for Metamath Proof Explorer - 22301-22400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnlmvscn 22301 The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 22304 and nlmtlm 22308. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    · = ( ·sf𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐾 = (TopOpen‘𝐹)       (𝑊 ∈ NrmMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))

Theoremrlmnlm 22302 The ring module over a normed ring is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod)

Theoremrlmnm 22303 The norm function in the ring module. (Contributed by AV, 9-Oct-2021.)
(norm‘𝑅) = (norm‘(ringLMod‘𝑅))

Theoremnrgtrg 22304 A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)

Theoremnrginvrcnlem 22305* Lemma for nrginvrcn 22306. Compare this proof with reccn2 14175, the elementary proof of continuity of division. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)    &   𝑁 = (norm‘𝑅)    &   𝐷 = (dist‘𝑅)    &   (𝜑𝑅 ∈ NrmRing)    &   (𝜑𝑅 ∈ NzRing)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵 ∈ ℝ+)    &   𝑇 = (if(1 ≤ ((𝑁𝐴) · 𝐵), 1, ((𝑁𝐴) · 𝐵)) · ((𝑁𝐴) / 2))       (𝜑 → ∃𝑥 ∈ ℝ+𝑦𝑈 ((𝐴𝐷𝑦) < 𝑥 → ((𝐼𝐴)𝐷(𝐼𝑦)) < 𝐵))

Theoremnrginvrcn 22306 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 = (invr𝑅)    &   𝐽 = (TopOpen‘𝑅)       (𝑅 ∈ NrmRing → 𝐼 ∈ ((𝐽t 𝑈) Cn (𝐽t 𝑈)))

Theoremnrgtdrg 22307 A normed division ring is a topological division ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ TopDRing)

Theoremnlmtlm 22308 A normed module is a topological module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(𝑊 ∈ NrmMod → 𝑊 ∈ TopMod)

Theoremisnvc 22309 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))

Theoremnvcnlm 22310 A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)

Theoremnvclvec 22311 A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec → 𝑊 ∈ LVec)

Theoremnvclmod 22312 A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec → 𝑊 ∈ LMod)

Theoremisnvc2 22313 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))

Theoremnvctvc 22314 A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec → 𝑊 ∈ TopVec)

Theoremlssnlm 22315 A subspace of a normed module is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ NrmMod ∧ 𝑈𝑆) → 𝑋 ∈ NrmMod)

Theoremlssnvc 22316 A subspace of a normed vector space is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ NrmVec ∧ 𝑈𝑆) → 𝑋 ∈ NrmVec)

Theoremrlmnvc 22317 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)

Theoremngpocelbl 22318 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)    &    + = (+g𝐺)    &   𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋))       ((𝐺 ∈ NrmMod ∧ 𝑅 ∈ ℝ* ∧ (𝑃𝑋𝐴𝑋)) → ((𝑃 + 𝐴) ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑁𝐴) < 𝑅))

12.4.9  Normed space homomorphisms (bounded linear operators)

Syntaxcnmo 22319 The operator norm function.
class normOp

Syntaxcnghm 22320 The class of normed group homomorphisms.
class NGHom

Syntaxcnmhm 22321 The class of normed module homomorphisms.
class NMHom

Definitiondf-nmo 22322* 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)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < )))

Definitiondf-nghm 22323* 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))

Definitiondf-nmhm 22324* 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)))

Theoremnmoffn 22325 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)

Theoremreldmnghm 22326 Lemma for normed group homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
Rel dom NGHom

Theoremreldmnmhm 22327 Lemma for module homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
Rel dom NMHom

Theoremnmofval 22328* 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[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )))

Theoremnmoval 22329* 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)) → (𝑁𝐹) = inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝐹𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < ))

Theoremnmogelb 22330* 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 ≤ (𝑁𝐹) ↔ ∀𝑟 ∈ (0[,)+∞)(∀𝑥𝑉 (𝑀‘(𝐹𝑥)) ≤ (𝑟 · (𝐿𝑥)) → 𝐴𝑟)))

Theoremnmolb 22331* 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) → (∀𝑥𝑉 (𝑀‘(𝐹𝑥)) ≤ (𝐴 · (𝐿𝑥)) → (𝑁𝐹) ≤ 𝐴))

Theoremnmolb2d 22332* 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 = (0g𝑆)    &   (𝜑𝑆 ∈ NrmGrp)    &   (𝜑𝑇 ∈ NrmGrp)    &   (𝜑𝐹 ∈ (𝑆 GrpHom 𝑇))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   ((𝜑 ∧ (𝑥𝑉𝑥0 )) → (𝑀‘(𝐹𝑥)) ≤ (𝐴 · (𝐿𝑥)))       (𝜑 → (𝑁𝐹) ≤ 𝐴)

Theoremnmof 22333 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*)

Theoremnmocl 22334 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)) → (𝑁𝐹) ∈ ℝ*)

Theoremnmoge0 22335 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 ≤ (𝑁𝐹))

Theoremnghmfval 22336 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)

Theoremisnghm 22337 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) ∧ (𝑁𝐹) ∈ ℝ)))

Theoremisnghm2 22338 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) ↔ (𝑁𝐹) ∈ ℝ))

Theoremisnghm3 22339 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) ↔ (𝑁𝐹) < +∞))

Theorembddnghm 22340 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 ∧ (𝑁𝐹) ≤ 𝐴)) → 𝐹 ∈ (𝑆 NGHom 𝑇))

Theoremnghmcl 22341 A normed group homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)       (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝑁𝐹) ∈ ℝ)

Theoremnmoi 22342 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) ∧ 𝑋𝑉) → (𝑀‘(𝐹𝑋)) ≤ ((𝑁𝐹) · (𝐿𝑋)))

Theoremnmoix 22343 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)) ∧ 𝑋𝑉) → (𝑀‘(𝐹𝑋)) ≤ ((𝑁𝐹) ·e (𝐿𝑋)))

Theoremnmoi2 22344 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 = (0g𝑆)       (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋𝑉𝑋0 )) → ((𝑀‘(𝐹𝑋)) / (𝐿𝑋)) ≤ (𝑁𝐹))

Theoremnmoleub 22345* 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 = (0g𝑆)    &   (𝜑𝑆 ∈ NrmGrp)    &   (𝜑𝑇 ∈ NrmGrp)    &   (𝜑𝐹 ∈ (𝑆 GrpHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 (𝑥0 → ((𝑀‘(𝐹𝑥)) / (𝐿𝑥)) ≤ 𝐴)))

Theoremnghmrcl1 22346 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)

Theoremnghmrcl2 22347 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)

Theoremnghmghm 22348 A normed group homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))

Theoremnmo0 22349 The operator norm of the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &    0 = (0g𝑇)       ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁‘(𝑉 × { 0 })) = 0)

Theoremnmoeq0 22350 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 = (0g𝑇)       ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → ((𝑁𝐹) = 0 ↔ 𝐹 = (𝑉 × { 0 })))

Theoremnmoco 22351 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‘(𝐹𝐺)) ≤ ((𝐿𝐹) · (𝑀𝐺)))

Theoremnghmco 22352 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)) → (𝐹𝐺) ∈ (𝑆 NGHom 𝑈))

Theoremnmotri 22353 Triangle inequality for the operator norm. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &    + = (+g𝑇)       ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹𝑓 + 𝐺)) ≤ ((𝑁𝐹) + (𝑁𝐺)))

Theoremnghmplusg 22354 The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
+ = (+g𝑇)       ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹𝑓 + 𝐺) ∈ (𝑆 NGHom 𝑇))

Theorem0nghm 22355 The zero operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
𝑉 = (Base‘𝑆)    &    0 = (0g𝑇)       ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇))

Theoremnmoid 22356 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 = (0g𝑆)       ((𝑆 ∈ NrmGrp ∧ { 0 } ⊊ 𝑉) → (𝑁‘( I ↾ 𝑉)) = 1)

Theoremidnghm 22357 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))

Theoremnmods 22358 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) ∧ 𝐴𝑉𝐵𝑉) → ((𝐹𝐴)𝐷(𝐹𝐵)) ≤ ((𝑁𝐹) · (𝐴𝐶𝐵)))

Theoremnghmcn 22359 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))

Theoremisnmhm 22360 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))))

Theoremnmhmrcl1 22361 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑆 ∈ NrmMod)

Theoremnmhmrcl2 22362 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑇 ∈ NrmMod)

Theoremnmhmlmhm 22363 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))

Theoremnmhmnghm 22364 A normed module homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 NGHom 𝑇))

Theoremnmhmghm 22365 A normed module homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))

Theoremisnmhm2 22366 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) ↔ (𝑁𝐹) ∈ ℝ))

Theoremnmhmcl 22367 A normed module homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)       (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑁𝐹) ∈ ℝ)

Theoremidnmhm 22368 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))

Theorem0nmhm 22369 The zero operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑉 = (Base‘𝑆)    &    0 = (0g𝑇)    &   𝐹 = (Scalar‘𝑆)    &   𝐺 = (Scalar‘𝑇)       ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇))

Theoremnmhmco 22370 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)) → (𝐹𝐺) ∈ (𝑆 NMHom 𝑈))

Theoremnmhmplusg 22371 The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
+ = (+g𝑇)       ((𝐹 ∈ (𝑆 NMHom 𝑇) ∧ 𝐺 ∈ (𝑆 NMHom 𝑇)) → (𝐹𝑓 + 𝐺) ∈ (𝑆 NMHom 𝑇))

12.4.10  Topology on the reals

Theoremqtopbaslem 22372 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

Theoremqtopbas 22373 The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)
((,) “ (ℚ × ℚ)) ∈ TopBases

Theoremretopbas 22374 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

Theoremretop 22375 The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
(topGen‘ran (,)) ∈ Top

Theoremuniretop 22376 The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
ℝ = (topGen‘ran (,))

Theoremretopon 22377 The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
(topGen‘ran (,)) ∈ (TopOn‘ℝ)

Theoremretps 22378 The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
𝐾 = {⟨(Base‘ndx), ℝ⟩, ⟨(TopSet‘ndx), (topGen‘ran (,))⟩}       𝐾 ∈ TopSp

Theoremiooretop 22379 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.)
(𝐴(,)𝐵) ∈ (topGen‘ran (,))

Theoremicccld 22380 Closed intervals are closed sets of the standard topology on . (Contributed by FL, 14-Sep-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))))

Theoremicopnfcld 22381 Right-unbounded closed intervals are closed sets of the standard topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
(𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))

Theoremiocmnfcld 22382 Left-unbounded closed intervals are closed sets of the standard topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
(𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))

Theoremqdensere 22383 is dense in the standard topology on . (Contributed by NM, 1-Mar-2007.)
((cls‘(topGen‘ran (,)))‘ℚ) = ℝ

Theoremcnmetdval 22384 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‘(𝐴𝐵)))

Theoremcnmet 22385 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)

Theoremcnxmet 22386 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
(abs ∘ − ) ∈ (∞Met‘ℂ)

Theoremcnbl0 22387 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))

Theoremcnblcld 22388* 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})

Theoremcnfldms 22389 The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld ∈ MetSp

Theoremcnfldxms 22390 The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld ∈ ∞MetSp

Theoremcnfldtps 22391 The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld ∈ TopSp

Theoremcnfldnm 22392 The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.)
abs = (norm‘ℂfld)

Theoremcnngp 22393 The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
fld ∈ NrmGrp

Theoremcnnrg 22394 The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
fld ∈ NrmRing

Theoremcnfldtopn 22395 The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 = (MetOpen‘(abs ∘ − ))

Theoremcnfldtopon 22396 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ (TopOn‘ℂ)

Theoremcnfldtop 22397 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Top

Theoremcnfldhaus 22398 The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Haus

Theoremzringnrg 22399 The ring of integers is a normed ring. (Contributed by AV, 13-Jun-2019.)
ring ∈ NrmRing

Theoremremetdval 22400 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‘(𝐴𝐵)))

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