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Theorem List for Metamath Proof Explorer - 22501-22600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtngtopn 22501 The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝐷 = (dist‘𝑇)    &   𝐽 = (MetOpen‘𝐷)       ((𝐺𝑉𝑁𝑊) → 𝐽 = (TopOpen‘𝑇))

Theoremtngnm 22502 The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &   𝐴 ∈ V       ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑁 = (norm‘𝑇))

Theoremtngngp2 22503 A norm turns a group into a normed group iff the generated metric is in fact a metric. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &   𝐷 = (dist‘𝑇)       (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))))

Theoremtngngpd 22504* Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑁:𝑋⟶ℝ)    &   ((𝜑𝑥𝑋) → ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))       (𝜑𝑇 ∈ NrmGrp)

Theoremtngngp 22505* Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &    0 = (0g𝐺)       (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))

Theoremtnggrpr 22506 If a structure equipped with a norm is a normed group, the structure itself must be a group. (Contributed by AV, 15-Oct-2021.)
𝑇 = (𝐺 toNrmGrp 𝑁)       ((𝑁𝑉𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)

Theoremtngngp3 22507* Alternate definition of a normed group (i.e. a group equipped with a norm) without using the properties of a metric space. This corresponds to the definition in N. H. Bingham, A. J. Ostaszewski: "Normed versus topological groups: dichotomy and duality", 2010, Dissertationes Mathematicae 472, pp. 1-138 and E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006. (Contributed by AV, 16-Oct-2021.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝐼 = (invg𝐺)       (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))

Theoremnrmtngdist 22508 The augmentation of a normed group by its own norm has the same distance function as the normed group (restricted to the base set). (Contributed by AV, 15-Oct-2021.)
𝑇 = (𝐺 toNrmGrp (norm‘𝐺))    &   𝑋 = (Base‘𝐺)       (𝐺 ∈ NrmGrp → (dist‘𝑇) = ((dist‘𝐺) ↾ (𝑋 × 𝑋)))

Theoremnrmtngnrm 22509 The augmentation of a normed group by its own norm is a normed group with the same norm. (Contributed by AV, 15-Oct-2021.)
𝑇 = (𝐺 toNrmGrp (norm‘𝐺))       (𝐺 ∈ NrmGrp → (𝑇 ∈ NrmGrp ∧ (norm‘𝑇) = (norm‘𝐺)))

Theoremtngngpim 22510 The induced metric of a normed group is a function. (Contributed by AV, 19-Oct-2021.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑁 = (norm‘𝐺)    &   𝑋 = (Base‘𝐺)    &   𝐷 = (dist‘𝑇)       (𝐺 ∈ NrmGrp → 𝐷:(𝑋 × 𝑋)⟶ℝ)

Theoremisnrg 22511 A normed ring is a ring with a norm that makes it into a normed group, and such that the norm is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (norm‘𝑅)    &   𝐴 = (AbsVal‘𝑅)       (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁𝐴))

Theoremnrgabv 22512 The norm of a normed ring is an absolute value. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (norm‘𝑅)    &   𝐴 = (AbsVal‘𝑅)       (𝑅 ∈ NrmRing → 𝑁𝐴)

Theoremnrgngp 22513 A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Theoremnrgring 22514 A normed ring is a ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ Ring)

Theoremnmmul 22515 The norm of a product in a normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑁 = (norm‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ NrmRing ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴 · 𝐵)) = ((𝑁𝐴) · (𝑁𝐵)))

Theoremnrgdsdi 22516 Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑁 = (norm‘𝑅)    &    · = (.r𝑅)    &   𝐷 = (dist‘𝑅)       ((𝑅 ∈ NrmRing ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝑁𝐴) · (𝐵𝐷𝐶)) = ((𝐴 · 𝐵)𝐷(𝐴 · 𝐶)))

Theoremnrgdsdir 22517 Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑁 = (norm‘𝑅)    &    · = (.r𝑅)    &   𝐷 = (dist‘𝑅)       ((𝑅 ∈ NrmRing ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵) · (𝑁𝐶)) = ((𝐴 · 𝐶)𝐷(𝐵 · 𝐶)))

Theoremnm1 22518 The norm of one in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑁 = (norm‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) → (𝑁1 ) = 1)

Theoremunitnmn0 22519 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 ∧ 𝐴𝑈) → (𝑁𝐴) ≠ 0)

Theoremnminvr 22520 The norm of an inverse in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑁 = (norm‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴𝑈) → (𝑁‘(𝐼𝐴)) = (1 / (𝑁𝐴)))

Theoremnmdvr 22521 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)    &    / = (/r𝑅)       (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴𝑋𝐵𝑈)) → (𝑁‘(𝐴 / 𝐵)) = ((𝑁𝐴) / (𝑁𝐵)))

Theoremnrgdomn 22522 A nonzero normed ring is a domain. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → (𝑅 ∈ Domn ↔ 𝑅 ∈ NzRing))

Theoremnrgtgp 22523 A normed ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp)

Theoremsubrgnrg 22524 A normed ring restricted to a subring is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐻 = (𝐺s 𝐴)       ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐻 ∈ NrmRing)

Theoremtngnrg 22525 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)       (𝐹𝐴𝑇 ∈ NrmRing)

Theoremisnlm 22526* 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)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐴 = (norm‘𝐹)       (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))

Theoremnmvs 22527 Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐴 = (norm‘𝐹)       ((𝑊 ∈ NrmMod ∧ 𝑋𝐾𝑌𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌)))

Theoremnlmngp 22528 A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

Theoremnlmlmod 22529 A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmMod → 𝑊 ∈ LMod)

Theoremnlmnrg 22530 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)

Theoremnlmngp2 22531 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)

Theoremnlmdsdi 22532 Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐷 = (dist‘𝑊)    &   𝐴 = (norm‘𝐹)       ((𝑊 ∈ NrmMod ∧ (𝑋𝐾𝑌𝑉𝑍𝑉)) → ((𝐴𝑋) · (𝑌𝐷𝑍)) = ((𝑋 · 𝑌)𝐷(𝑋 · 𝑍)))

Theoremnlmdsdir 22533 Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐷 = (dist‘𝑊)    &   𝑁 = (norm‘𝑊)    &   𝐸 = (dist‘𝐹)       ((𝑊 ∈ NrmMod ∧ (𝑋𝐾𝑌𝐾𝑍𝑉)) → ((𝑋𝐸𝑌) · (𝑁𝑍)) = ((𝑋 · 𝑍)𝐷(𝑌 · 𝑍)))

Theoremnlmmul0or 22534 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)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑂 = (0g𝐹)       ((𝑊 ∈ NrmMod ∧ 𝐴𝐾𝐵𝑉) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 𝑂𝐵 = 0 )))

Theoremsranlm 22535 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)

Theoremnlmvscnlem2 22536 Lemma for nlmvscn 22538. Compare this proof with the similar elementary proof mulcn2 14370 for continuity of multiplication on . (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)    &    · = ( ·𝑠𝑊)    &   𝑇 = ((𝑅 / 2) / ((𝐴𝐵) + 1))    &   𝑈 = ((𝑅 / 2) / ((𝑁𝑋) + 𝑇))    &   (𝜑𝑊 ∈ NrmMod)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝐶𝐾)    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝐵𝐸𝐶) < 𝑈)    &   (𝜑 → (𝑋𝐷𝑌) < 𝑇)       (𝜑 → ((𝐵 · 𝑋)𝐷(𝐶 · 𝑌)) < 𝑅)

Theoremnlmvscnlem1 22537* Lemma for nlmvscn 22538. (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)    &    · = ( ·𝑠𝑊)    &   𝑇 = ((𝑅 / 2) / ((𝐴𝐵) + 1))    &   𝑈 = ((𝑅 / 2) / ((𝑁𝑋) + 𝑇))    &   (𝜑𝑊 ∈ NrmMod)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → ∃𝑟 ∈ ℝ+𝑥𝐾𝑦𝑉 (((𝐵𝐸𝑥) < 𝑟 ∧ (𝑋𝐷𝑦) < 𝑟) → ((𝐵 · 𝑋)𝐷(𝑥 · 𝑦)) < 𝑅))

Theoremnlmvscn 22538 The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 22541 and nlmtlm 22545. (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 22539 The ring module over a normed ring is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod)

Theoremrlmnm 22540 The norm function in the ring module. (Contributed by AV, 9-Oct-2021.)
(norm‘𝑅) = (norm‘(ringLMod‘𝑅))

Theoremnrgtrg 22541 A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)

Theoremnrginvrcnlem 22542* Lemma for nrginvrcn 22543. Compare this proof with reccn2 14371, 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 22543 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 22544 A normed division ring is a topological division ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ TopDRing)

Theoremnlmtlm 22545 A normed module is a topological module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(𝑊 ∈ NrmMod → 𝑊 ∈ TopMod)

Theoremisnvc 22546 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 22547 A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)

Theoremnvclvec 22548 A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec → 𝑊 ∈ LVec)

Theoremnvclmod 22549 A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec → 𝑊 ∈ LMod)

Theoremisnvc2 22550 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 22551 A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec → 𝑊 ∈ TopVec)

Theoremlssnlm 22552 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 22553 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 22554 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 22555 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 22556 The operator norm function.
class normOp

Syntaxcnghm 22557 The class of normed group homomorphisms.
class NGHom

Syntaxcnmhm 22558 The class of normed module homomorphisms.
class NMHom

Definitiondf-nmo 22559* 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 22560* 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 22561* 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 22562 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 22563 Lemma for normed group homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
Rel dom NGHom

Theoremreldmnmhm 22564 Lemma for module homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
Rel dom NMHom

Theoremnmofval 22565* 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 22566* 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 22567* 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 22568* 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 22569* 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 22570 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 22571 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 22572 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 22573 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 22574 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 22575 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 22576 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 22577 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 22578 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 22579 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 22580 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 22581 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 22582* 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 22583 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)

Theoremnghmrcl2 22584 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)

Theoremnghmghm 22585 A normed group homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))

Theoremnmo0 22586 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 22587 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 22588 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 22589 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 22590 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 22591 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 22592 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 22593 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 22594 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 22595 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 22596 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 22597 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 22598 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑆 ∈ NrmMod)

Theoremnmhmrcl2 22599 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑇 ∈ NrmMod)

Theoremnmhmlmhm 22600 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))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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