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Theorem List for Metamath Proof Explorer - 18701-18800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnmrtri 18701 Reverse triangle inequality for the norm of a subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremnm2dif 18702 Inequality for the difference of norms. (Contributed by Mario Carneiro, 6-Oct-2015.)
NrmGrp

Theoremnmtri 18703 The triangle inequality for the norm of a sum. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremnm0 18704 Norm of the identity element. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremsubgnm 18705 The norm in a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.)
s                      SubGrp

Theoremsubgnm2 18706 A substructure assigns the same values to the norms of elements of a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.)
s                      SubGrp

Theoremsubgngp 18707 A normed group restricted to a subgroup is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
s        NrmGrp SubGrp NrmGrp

Theoremngptgp 18708 A normed abelian group is a topological group (with the topology induced by the metric induced by the norm). (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremngppropd 18709* Property deduction for a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp NrmGrp

Theoremreldmtng 18710 The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.)
toNrmGrp

Theoremtngval 18711 Value of the function which augments a given structure with a norm . (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp                             sSet sSet TopSet

Theoremtnglem 18712 Lemma for tngbas 18713 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp        Slot

Theoremtngbas 18713 The base set of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp

Theoremtngplusg 18714 The group addition of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp

Theoremtng0 18715 The group identity of a structure augmented with a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
toNrmGrp

Theoremtngmulr 18716 The ring multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp

Theoremtngsca 18717 The scalar ring of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp        Scalar       Scalar

Theoremtngvsca 18718 The scalar multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp

Theoremtngip 18719 The inner product operation of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp

Theoremtngds 18720 The metric function of a structure augmented with a norm. (Contributed by Mario Carneiro, 3-Oct-2015.)
toNrmGrp

Theoremtngtset 18721 The topology generated by a normed structure. (Contributed by Mario Carneiro, 3-Oct-2015.)
toNrmGrp                      TopSet

Theoremtngtopn 18722 The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
toNrmGrp

Theoremtngnm 18723 The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
toNrmGrp

Theoremtngngp2 18724 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.)
toNrmGrp                      NrmGrp

Theoremtngngpd 18725* Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
toNrmGrp                                                         NrmGrp

Theoremtngngp 18726* Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
toNrmGrp                             NrmGrp

Theoremisnrg 18727 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.)
AbsVal       NrmRing NrmGrp

Theoremnrgabv 18728 The norm of a normed ring is an absolute value. (Contributed by Mario Carneiro, 4-Oct-2015.)
AbsVal       NrmRing

Theoremnrgngp 18729 A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing NrmGrp

Theoremnrgrng 18730 A normed ring is a ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing

Theoremnmmul 18731 The norm of a product in a normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
NrmRing

Theoremnrgdsdi 18732 Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
NrmRing

Theoremnrgdsdir 18733 Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
NrmRing

Theoremnm1 18734 The norm of one in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
NrmRing NzRing

Theoremunitnmn0 18735 The norm of a unit is nonzero in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Unit       NrmRing NzRing

Theoremnminvr 18736 The norm of an inverse in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Unit              NrmRing NzRing

Theoremnmdvr 18737 The norm of a division in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Unit       /r       NrmRing NzRing

Theoremnrgdomn 18738 A nonzero normed ring is a domain. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing Domn NzRing

Theoremnrgtgp 18739 A normed ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
NrmRing

Theoremsubrgnrg 18740 A normed ring restricted to a subring is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
s        NrmRing SubRing NrmRing

Theoremtngnrg 18741 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.)
toNrmGrp        AbsVal       NrmRing

Theoremisnlm 18742* 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.)
Scalar                     NrmMod NrmGrp NrmRing

Theoremnmvs 18743 Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Scalar                     NrmMod

Theoremnlmngp 18744 A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmMod NrmGrp

Theoremnlmlmod 18745 A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmMod

Theoremnlmnrg 18746 The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
Scalar       NrmMod NrmRing

Theoremnlmngp2 18747 The scalar component of a left module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
Scalar       NrmMod NrmGrp

Theoremnlmdsdi 18748 Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
Scalar                            NrmMod

Theoremnlmdsdir 18749 Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
Scalar                                   NrmMod

Theoremnlmmul0or 18750 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.)
Scalar                     NrmMod

Theoremsranlm 18751 The subring algebra over a normed ring is a normed left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg        NrmRing SubRing NrmMod

Theoremnlmvscnlem2 18752 Lemma for nlmvscn 18754. Compare this proof with the similar elementary proof mulcn2 12420 for continuity of multiplication on . (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                                                                      NrmMod

Theoremnlmvscnlem1 18753* Lemma for nlmvscn 18754. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                                                                      NrmMod

Theoremnlmvscn 18754 The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 18756 and nlmtlm 18760. (Contributed by Mario Carneiro, 4-Oct-2015.)
Scalar                            NrmMod

Theoremrlmnlm 18755 The ring module over a normed ring is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing ringLMod NrmMod

Theoremnrgtrg 18756 A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing

Theoremnrginvrcnlem 18757* Lemma for nrginvrcn 18758. Compare this proof with reccn2 12421, the elementary proof of continuity of division. (Contributed by Mario Carneiro, 6-Oct-2015.)
Unit                            NrmRing       NzRing

Theoremnrginvrcn 18758 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.)
Unit                     NrmRing t t

Theoremnrgtdrg 18759 A normed division ring is a topological division ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
NrmRing TopDRing

Theoremnlmtlm 18760 A normed module is a topological module. (Contributed by Mario Carneiro, 6-Oct-2015.)
NrmMod TopMod

Theoremisnvc 18761 A normed vector space is just a normed module which is algebraically a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec NrmMod

Theoremnvcnlm 18762 A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec NrmMod

Theoremnvclvec 18763 A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec

Theoremnvclmod 18764 A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec

Theoremisnvc2 18765 A normed vector space is just a normed module whose scalar ring is a division ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
Scalar       NrmVec NrmMod

Theoremnvctvc 18766 A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec

Theoremlssnlm 18767 A subspace of a normed module is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
s               NrmMod NrmMod

Theoremlssnvc 18768 A subspace of a normed vector space is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
s               NrmVec NrmVec

Theoremrlmnvc 18769 The ring module over a normed division ring is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing ringLMod NrmVec

11.4.9  Normed space homomorphisms (bounded linear operators)

Syntaxcnmo 18770 The operator norm function.

Syntaxcnghm 18771 The class of normed group homomorphisms.
NGHom

Syntaxcnmhm 18772 The class of normed module homomorphisms.
NMHom

Definitiondf-nmo 18773* 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 . Equivalent to the definition of linear operator norm in [AkhiezerGlazman] p. 39. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Definitiondf-nghm 18774* 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 NrmGrp NrmGrp

Definitiondf-nmhm 18775* 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 such that (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom NrmMod NrmMod LMHom NGHom

Theoremnmoffn 18776 The function producing operator norm functions is a function on normed groups. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremreldmnghm 18777 Lemma for normed group homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremreldmnmhm 18778 Lemma for module homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom

Theoremnmofval 18779* Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoval 18780* Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmogelb 18781* Property of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmolb 18782* 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.)
NrmGrp NrmGrp

Theoremnmolb2d 18783* 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.)
NrmGrp       NrmGrp

Theoremnmof 18784 The operator norm is a function into the extended reals. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmocl 18785 The operator norm of an operator is an extended real. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoge0 18786 The operator norm of an operator is nonnegative. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnghmfval 18787 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremisnghm 18788 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom NrmGrp NrmGrp

Theoremisnghm2 18789 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theoremisnghm3 18790 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theorembddnghm 18791 A bounded group homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theoremnghmcl 18792 A normed group homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremnmoi 18793 The operator norm achieves the minimum of the set of upper bounds, if the operator is bounded. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremnmoix 18794 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.)
NrmGrp NrmGrp

Theoremnmoi2 18795 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.)
NrmGrp NrmGrp

Theoremnmoleub 18796* The operator norm, defined as an infimum of upper bounds, can also be defined as a supremum of norms of away from zero. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp       NrmGrp

Theoremnghmrcl1 18797 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom NrmGrp

Theoremnghmrcl2 18798 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom NrmGrp

Theoremnghmghm 18799 A normed group homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremnmo0 18800 The operator norm of the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
NrmGrp NrmGrp

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