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Theorem List for Metamath Proof Explorer - 18201-18300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnrginvrcnlem 18201* Lemma for nrginvrcn 18202. Compare this proof with reccn2 12070, the elementary proof of continuity of division. (Contributed by Mario Carneiro, 6-Oct-2015.)
Unit                            NrmRing       NzRing

Theoremnrginvrcn 18202 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 18203 A normed division ring is a topological division ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
NrmRing TopDRing

Theoremnlmtlm 18204 A normed module is a topological module. (Contributed by Mario Carneiro, 6-Oct-2015.)
NrmMod TopMod

Theoremisnvc 18205 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 18206 A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec NrmMod

Theoremnvclvec 18207 A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec

Theoremnvclmod 18208 A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec

Theoremisnvc2 18209 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 18210 A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec

Theoremlssnlm 18211 A subspace of a normed module is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
s               NrmMod NrmMod

Theoremlssnvc 18212 A subspace of a normed vector space is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
s               NrmVec NrmVec

Theoremrlmnvc 18213 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 18214 The operator norm function.

Syntaxcnghm 18215 The class of normed group homomorphisms.
NGHom

Syntaxcnmhm 18216 The class of normed module homomorphisms.
NMHom

Definitiondf-nmo 18217* 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 18218* 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 18219* 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 18220 The function producing operator norm functions is a function on normed groups. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremreldmnghm 18221 Lemma for normed group homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremreldmnmhm 18222 Lemma for module homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom

Theoremnmofval 18223* Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoval 18224* Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmogelb 18225* Property of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmolb 18226* 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 18227* 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 18228 The operator norm is a function into the extended reals. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmocl 18229 The operator norm of an operator is an extended real. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoge0 18230 The operator norm of an operator is nonnegative. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp

Theoremnghmfval 18231 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremisnghm 18232 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom NrmGrp NrmGrp

Theoremisnghm2 18233 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theoremisnghm3 18234 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theorembddnghm 18235 A bounded group homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theoremnghmcl 18236 A normed group homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremnmoi 18237 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 18238 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 18239 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 18240* 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 18241 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom NrmGrp

Theoremnghmrcl2 18242 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom NrmGrp

Theoremnghmghm 18243 A normed group homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom

Theoremnmo0 18244 The operator norm of the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoeq0 18245 The operator norm is zero only for the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
NrmGrp NrmGrp

Theoremnmoco 18246 An upper bound on the operator norm of a composition. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom NGHom

Theoremnghmco 18247 The composition of normed group homomorphisms is a normed group homomorphism. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom NGHom NGHom

Theoremnmotri 18248 Triangle inequality for the operator norm. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom NGHom

Theoremnghmplusg 18249 The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom NGHom NGHom

Theorem0nghm 18250 The zero operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NrmGrp NGHom

Theoremnmoid 18251 The operator norm of the identity function on a nontrivial group. (Contributed by Mario Carneiro, 20-Oct-2015.)
NrmGrp

Theoremidnghm 18252 The identity operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NrmGrp NGHom

Theoremnmods 18253 Upper bound for the distance between the values of a bounded linear operator. (Contributed by Mario Carneiro, 22-Oct-2015.)
NGHom

Theoremnghmcn 18254 A normed group homomorphism is a continuous function. (Contributed by Mario Carneiro, 20-Oct-2015.)
NGHom

Theoremisnmhm 18255 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 18256 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom NrmMod

Theoremnmhmrcl2 18257 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom NrmMod

Theoremnmhmlmhm 18258 A normed module homomorphism is a left module homomorphism (a linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom LMHom

Theoremnmhmnghm 18259 A normed module homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom NGHom

Theoremnmhmghm 18260 A normed module homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom

Theoremisnmhm2 18261 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 18262 A normed module homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom

Theoremidnmhm 18263 The identity operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
NrmMod NMHom

Theorem0nmhm 18264 The zero operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
Scalar       Scalar       NrmMod NrmMod NMHom

Theoremnmhmco 18265 The composition of bounded linear operators is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
NMHom NMHom NMHom

Theoremnmhmplusg 18266 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 18267 The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremqtopbas 18268 The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)

Theoremretopbas 18269 A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)

Theoremretop 18270 The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)

Theoremuniretop 18271 The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)

Theoremretopon 18272 The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
TopOn

TheoremretpsOLD 18273 The standard topological space on the reals. (Contributed by NM, 10-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremretps 18274 The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
TopSet

Theoremiooretop 18275 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.)

Theoremicccld 18276 Closed intervals are closed sets of the standard topology on . (Contributed by FL, 14-Sep-2007.)

Theoremicopnfcld 18277 Right-unbounded closed intervals are closed sets of the standard topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)

Theoremiocmnfcld 18278 Left-unbounded closed intervals are closed sets of the standard topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)

Theoremqdensere 18279 is dense in the standard topology on . (Contributed by NM, 1-Mar-2007.)

Theoremcnmetdval 18280 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.)

Theoremcnmet 18281 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.)

Theoremcnxmet 18282 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)

Theoremcnbl0 18283 Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremcnblcld 18284* Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremcnfldms 18285 The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld

Theoremcnfldxms 18286 The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld

Theoremcnfldtps 18287 The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld

Theoremcnfldnm 18288 The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.)
fld

Theoremcnngp 18289 The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
fld NrmGrp

Theoremcnnrg 18290 The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
fld NrmRing

Theoremcnfldtopn 18291 The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld

Theoremcnfldtopon 18292 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
fld       TopOn

Theoremcnfldtop 18293 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
fld

Theoremcnfldhaus 18294 The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.)
fld

Theoremremetdval 18295 Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.)

Theoremremet 18296 The absolute value metric determines a metric space on the reals. (Contributed by NM, 10-Feb-2007.)

Theoremrexmet 18297 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)

Theorembl2ioo 18298 A ball in terms of an open interval of reals. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremioo2bl 18299 An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

Theoremioo2blex 18300 An open interval of reals in terms of a ball. (Contributed by Mario Carneiro, 14-Nov-2013.)

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