P. 247 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nvclvec 24601	A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LVec)
Theorem	nvclmod 24602	A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LMod)
Theorem	isnvc2 24603	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 24604	A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ TopVec)
Theorem	lssnlm 24605	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 24606	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 24607	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 24608	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))
12.4.9  Normed space homomorphisms (bounded linear operators)
Syntax	cnmo 24609	The operator norm function.
class normOp
Syntax	cnghm 24610	The class of normed group homomorphisms.
class NGHom
Syntax	cnmhm 24611	The class of normed module homomorphisms.
class NMHom
Definition	df-nmo 24612*	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 24613*	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 24614*	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 24615	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 24616	Lemma for normed group homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ Rel dom NGHom
Theorem	reldmnmhm 24617	Lemma for module homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ Rel dom NMHom
Theorem	nmofval 24618*	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 24619*	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 24620*	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 24621*	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 24622*	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 24623	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 24624	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 24625	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 24626	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 24627	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 24628	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 24629	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 24630	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 24631	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 24632	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 24633	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 24634	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 24635*	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 24636	Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
Theorem	nghmrcl2 24637	Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
Theorem	nghmghm 24638	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 24639	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 24640	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 24641	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 24642	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 24643	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 24644	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 24645	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 24646	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 24647	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 24648	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 24649	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 24650	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 24651	Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑆 ∈ NrmMod)
Theorem	nmhmrcl2 24652	Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑇 ∈ NrmMod)
Theorem	nmhmlmhm 24653	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 24654	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 24655	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 24656	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 24657	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 24658	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 24659	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 24660	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 24661	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))
12.4.10  Topology on the reals
Theorem	qtopbaslem 24662	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 24663	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 24664	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 24665	The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
⊢ (topGen‘ran (,)) ∈ Top
Theorem	uniretop 24666	The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
⊢ ℝ = ∪ (topGen‘ran (,))
Theorem	retopon 24667	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 24668	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 24669	Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.)
⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
Theorem	icccld 24670	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 24671	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 24672	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 24673	ℚ is dense in the standard topology on ℝ. (Contributed by NM, 1-Mar-2007.)
⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ
Theorem	cnmetdval 24674	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 24675	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 24676	The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
Theorem	cnbl0 24677	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 24678*	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 24679	The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ MetSp
Theorem	cnfldxms 24680	The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ ∞MetSp
Theorem	cnfldtps 24681	The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ TopSp
Theorem	cnfldnm 24682	The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ abs = (norm‘ℂfld)
Theorem	cnngp 24683	The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ ℂfld ∈ NrmGrp
Theorem	cnnrg 24684	The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ ℂfld ∈ NrmRing
Theorem	cnfldtopn 24685	The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
Theorem	cnfldtopon 24686	The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ (TopOn‘ℂ)
Theorem	cnfldtop 24687	The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ Top
Theorem	cnfldhaus 24688	The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ Haus
Theorem	unicntop 24689	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 24690	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	cnn0opn 24691	The set of nonzero complex numbers is open with respect to the standard topology on complex numbers. (Contributed by SN, 7-Oct-2025.)
⊢ (ℂ ∖ {0}) ∈ (TopOpen‘ℂfld)
Theorem	zringnrg 24692	The ring of integers is a normed ring. (Contributed by AV, 13-Jun-2019.)
⊢ ℤring ∈ NrmRing
Theorem	remetdval 24693	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 24694	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 24695	The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ 𝐷 ∈ (∞Met‘ℝ)
Theorem	bl2ioo 24696	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 24697	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 24698	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 24699	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 24700	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