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Theorem List for Metamath Proof Explorer - 18001-18100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtngngpd 18001* Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
toNrmGrp                                                         NrmGrp

Theoremtngngp 18002* Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
toNrmGrp                             NrmGrp

Theoremisnrg 18003 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 18004 The norm of a normed ring is an absolute value. (Contributed by Mario Carneiro, 4-Oct-2015.)
AbsVal       NrmRing

Theoremnrgngp 18005 A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing NrmGrp

Theoremnrgrng 18006 A normed ring is a ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing

Theoremnmmul 18007 The norm of a product in a normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
NrmRing

Theoremnrgdsdi 18008 Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
NrmRing

Theoremnrgdsdir 18009 Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
NrmRing

Theoremnm1 18010 The norm of one in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
NrmRing NzRing

Theoremunitnmn0 18011 The norm of a unit is nonzero in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Unit       NrmRing NzRing

Theoremnminvr 18012 The norm of an inverse in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Unit              NrmRing NzRing

Theoremnmdvr 18013 The norm of a division in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Unit       /r       NrmRing NzRing

Theoremnrgdomn 18014 A nonzero normed ring is a domain. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing Domn NzRing

Theoremnrgtgp 18015 A normed ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
NrmRing

Theoremsubrgnrg 18016 A normed ring restricted to a subring is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
s        NrmRing SubRing NrmRing

Theoremtngnrg 18017 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 18018* 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 18019 Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Scalar                     NrmMod

Theoremnlmngp 18020 A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmMod NrmGrp

Theoremnlmlmod 18021 A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmMod

Theoremnlmnrg 18022 The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
Scalar       NrmMod NrmRing

Theoremnlmngp2 18023 The scalar component of a left module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
Scalar       NrmMod NrmGrp

Theoremnlmdsdi 18024 Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
Scalar                            NrmMod

Theoremnlmdsdir 18025 Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
Scalar                                   NrmMod

Theoremnlmmul0or 18026 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 18027 The subring algebra over a normed ring is a normed left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg        NrmRing SubRing NrmMod

Theoremnlmvscnlem2 18028 Lemma for nlmvscn 18030. Compare this proof with the similar elementary proof mulcn2 11946 for continuity of multiplication on . (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                                                                      NrmMod

Theoremnlmvscnlem1 18029* Lemma for nlmvscn 18030. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                                                                      NrmMod

Theoremnlmvscn 18030 The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 18032 and nlmtlm 18036. (Contributed by Mario Carneiro, 4-Oct-2015.)
Scalar                            NrmMod

Theoremrlmnlm 18031 The ring module over a normed ring is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing ringLMod NrmMod

Theoremnrgtrg 18032 A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing

Theoremnrginvrcnlem 18033* Lemma for nrginvrcn 18034. Compare this proof with reccn2 11947, the elementary proof of continuity of division. (Contributed by Mario Carneiro, 6-Oct-2015.)
Unit                            NrmRing       NzRing

Theoremnrginvrcn 18034 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 18035 A normed division ring is a topological division ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
NrmRing TopDRing

Theoremnlmtlm 18036 A normed module is a topological module. (Contributed by Mario Carneiro, 6-Oct-2015.)
NrmMod TopMod

Theoremisnvc 18037 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 18038 A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec NrmMod

Theoremnvclvec 18039 A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec

Theoremnvclmod 18040 A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec

Theoremisnvc2 18041 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 18042 A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec

Theoremlssnlm 18043 A subspace of a normed module is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
s               NrmMod NrmMod

Theoremlssnvc 18044 A subspace of a normed vector space is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
s               NrmVec NrmVec

Theoremrlmnvc 18045 The ring module over a normed division ring is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing ringLMod NrmVec

11.3.7  Normed space homomorphisms (bounded linear operators)

Syntaxcnmo 18046 The operator norm function.

Syntaxcnghm 18047 The class of normed group homomorphisms.
NGHom

Syntaxcnmhm 18048 The class of normed module homomorphisms.
NMHom

Definitiondf-nmo 18049* 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 18050* 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 18051* 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 18052 The function producing operator norm functions is a function on normed groups. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremreldmnghm 18053 Lemma for normed group homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremreldmnmhm 18054 Lemma for module homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom

Theoremnmofval 18055* Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoval 18056* Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmogelb 18057* Property of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmolb 18058* 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 18059* 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 18060 The operator norm is a function into the extended reals. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmocl 18061 The operator norm of an operator is an extended real. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoge0 18062 The operator norm of an operator is nonnegative. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnghmfval 18063 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremisnghm 18064 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom NrmGrp NrmGrp

Theoremisnghm2 18065 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theoremisnghm3 18066 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theorembddnghm 18067 A bounded group homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theoremnghmcl 18068 A normed group homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremnmoi 18069 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 18070 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 18071 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 18072* 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 18073 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom NrmGrp

Theoremnghmrcl2 18074 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom NrmGrp

Theoremnghmghm 18075 A normed group homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremnmo0 18076 The operator norm of the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoeq0 18077 The operator norm is zero only for the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoco 18078 An upper bound on the operator norm of a composition. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom NGHom

Theoremnghmco 18079 The composition of normed group homomorphisms is a normed group homomorphism. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom NGHom NGHom

Theoremnmotri 18080 Triangle inequality for the operator norm. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom NGHom

Theoremnghmplusg 18081 The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom NGHom NGHom

Theorem0nghm 18082 The zero operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theoremnmoid 18083 The operator norm of the identity function on a nontrivial group. (Contributed by Mario Carneiro, 20-Oct-2015.)
NrmGrp

Theoremidnghm 18084 The identity operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NGHom

Theoremnmods 18085 Upper bound for the distance between the values of a bounded linear operator. (Contributed by Mario Carneiro, 22-Oct-2015.)
NGHom

Theoremnghmcn 18086 A normed group homomorphism is a continuous function. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom

Theoremisnmhm 18087 A normed module homomorphism is a left module homomorphism which is also a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom NrmMod NrmMod LMHom NGHom

Theoremnmhmrcl1 18088 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom NrmMod

Theoremnmhmrcl2 18089 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom NrmMod

Theoremnmhmlmhm 18090 A normed module homomorphism is a left module homomorphism (a linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom LMHom

Theoremnmhmnghm 18091 A normed module homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom NGHom

Theoremnmhmghm 18092 A normed module homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom

Theoremisnmhm2 18093 A normed module homomorphism is a left module homomorphism with bounded norm (a bounded linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmMod NrmMod LMHom NMHom

Theoremnmhmcl 18094 A normed module homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom

Theoremidnmhm 18095 The identity operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
NrmMod NMHom

Theorem0nmhm 18096 The zero operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
Scalar       Scalar       NrmMod NrmMod NMHom

Theoremnmhmco 18097 The composition of bounded linear operators is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
NMHom NMHom NMHom

Theoremnmhmplusg 18098 The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
NMHom NMHom NMHom

11.3.8  Topology on the Reals

Theoremqtopbaslem 18099 The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremqtopbas 18100 The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)

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