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Theorem List for Metamath Proof Explorer - 19201-19300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcfilresi 19201 A Cauchy filter on a metric subspace extends to a Cauchy filter in the larger space. (Contributed by Mario Carneiro, 15-Oct-2015.)
CauFil CauFil

Theoremcfilres 19202 Cauchy filter on a metric subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
CauFil t CauFil

Theoremcaussi 19203 Cauchy sequence on a metric subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)

Theoremcauss 19204 Cauchy sequence on a metric subspace. (Contributed by NM, 29-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)

Theoremequivcfil 19205* If the metric is "strongly finer" than (meaning that there is a positive real constant such that ), all the -Cauchy filters are also -Cauchy. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they have the same Cauchy sequences.) (Contributed by Mario Carneiro, 14-Sep-2015.)
CauFil CauFil

Theoremequivcau 19206* If the metric is "strongly finer" than (meaning that there is a positive real constant such that ), all the -Cauchy sequences are also -Cauchy. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they have the same Cauchy sequences.) (Contributed by Mario Carneiro, 14-Sep-2015.)

Theoremlmle 19207* If the distance from each member of a converging sequence to a given point is less than or equal to a given amount, so is the convergence value. (Contributed by NM, 23-Dec-2007.) (Proof shortened by Mario Carneiro, 1-May-2014.)

Theoremlmclim 19208 Relate a limit on the metric space of complex numbers to our complex number limit notation. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
fld

Theoremlmclimf 19209 Relate a limit on the metric space of complex numbers to our complex number limit notation. (Contributed by NM, 24-Jul-2007.) (Revised by Mario Carneiro, 1-May-2014.)
fld

Theoremmetelcls 19210* A point belongs to the closure of a subset iff there is a sequence in the subset converging to it. Theorem 1.4-6(a) of [Kreyszig] p. 30. This proof uses countable choice ax-cc 8271. The statement can be generalized to first-countable spaces, not just metrizable spaces. (Contributed by NM, 8-Nov-2007.) (Proof shortened by Mario Carneiro, 1-May-2015.)

Theoremmetcld 19211* A subset of a metric space is closed iff every convergent sequence on it converges to a point in the subset. Theorem 1.4-6(b) of [Kreyszig] p. 30. (Contributed by NM, 11-Nov-2007.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremmetcld2 19212 A subset of a metric space is closed iff every convergent sequence on it converges to a point in the subset. Theorem 1.4-6(b) of [Kreyszig] p. 30. (Contributed by Mario Carneiro, 1-May-2014.)

Theoremcaubl 19213* Sufficient condition to ensure a sequence of nested balls is Cauchy. (Contributed by Mario Carneiro, 18-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremcaublcls 19214* The convergent point of a sequence of nested balls is in the closures of any of the balls (i.e. it is in the intersection of the closures). Indeed, it is the only point in the intersection because a metric space is Hausdorff, but we don't prove this here. (Contributed by Mario Carneiro, 21-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremmetcnp4 19215* Two ways to say a mapping from metric to metric is continuous at point . Theorem 14-4.3 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 4-May-2014.)

Theoremmetcn4 19216* Two ways to say a mapping from metric to metric is continuous. Theorem 10.3 of [Munkres] p. 128. (Contributed by NM, 13-Jun-2007.) (Revised by Mario Carneiro, 4-May-2014.)

Theoremiscmet3i 19217* Properties that determine a complete metric space. (Contributed by NM, 15-Apr-2007.) (Revised by Mario Carneiro, 5-May-2014.)

Theoremlmcau 19218 Every convergent sequence in a metric space is a Cauchy sequence. Theorem 1.4-5 of [Kreyszig] p. 28. (Contributed by NM, 29-Jan-2008.) (Proof shortened by Mario Carneiro, 5-May-2014.)

Theoremflimcfil 19219 Every convergent filter in a metric space is a Cauchy filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
CauFil

Theoremcmetss 19220 A subspace of a complete metric space is complete iff it is closed in the parent space. Theorem 1.4-7 of [Kreyszig] p. 30. (Contributed by NM, 28-Jan-2008.) (Revised by Mario Carneiro, 15-Oct-2015.)

Theoremequivcmet 19221* If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 19206, metss2 18495, this theorem does not have a one-directional form - it is possible for a metric that is strongly finer than the complete metric to be incomplete and vice versa. Consider the metric on induced by the usual homeomorphism from against the usual metric on and against the discrete metric on . Then both and are complete but is not, and is strongly finer than , which is strongly finer than . (Contributed by Mario Carneiro, 15-Sep-2015.)

Theoremrelcmpcmet 19222* If is a metric space such that all the balls of some fixed size are relatively compact, then is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)
t

Theoremcmpcmet 19223 A compact metric space is complete. One half of heibor 26420. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremcfilucfil3OLD 19224 Given a metric and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the Cauchy filters for the metric. (Contributed by Thierry Arnoux, 15-Dec-2017.)
CauFilumetUnifOLD CauFil

Theoremcfilucfil3 19225 Given a metric and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the Cauchy filters for the metric. (Contributed by Thierry Arnoux, 15-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
CauFilumetUnif CauFil

Theoremcfilucfil4OLD 19226 Given a metric and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the Cauchy filters for the metric. (Contributed by Thierry Arnoux, 15-Dec-2017.)
CauFilumetUnifOLD CauFil

Theoremcfilucfil4 19227 Given a metric and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the Cauchy filters for the metric. (Contributed by Thierry Arnoux, 15-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
CauFilumetUnif CauFil

Theoremcncmet 19228 The set of complex numbers is a complete metric space under the absolute value metric. (Contributed by NM, 20-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2015.)

Theoremrecmet 19229 The real numbers are a complete metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

11.5.4  Baire's Category Theorem

Theorembcthlem1 19230* Lemma for bcth 19235. Substitutions for the function . (Contributed by Mario Carneiro, 9-Jan-2014.)

Theorembcthlem2 19231* Lemma for bcth 19235. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)

Theorembcthlem3 19232* Lemma for bcth 19235. The limit point of the centers in the sequence is in the intersection of every ball in the sequence. (Contributed by Mario Carneiro, 7-Jan-2014.)

Theorembcthlem4 19233* Lemma for bcth 19235. Given any open ball as starting point (and in particular, a ball in ), the limit point of the centers of the induced sequence of balls is outside . Note that a set has empty interior iff every nonempty open set contains points outside , i.e. . (Contributed by Mario Carneiro, 7-Jan-2014.)

Theorembcthlem5 19234* Lemma for bcth 19235. The proof makes essential use of the Axiom of Dependent Choice axdc4uz 11277, which in the form used here accepts a "selection" function from each element of to a nonempty subset of , and the result function maps to an element of . The trick here is thus in the choice of and : we let be the set of all tagged nonempty open sets (tagged here meaning that we have a point and an open set, in an ordered pair), and gives the set of all balls of size less than , tagged by their centers, whose closures fit within the given open set and miss .

Since is closed, is open and also nonempty, since is nonempty and has empty interior. Then there is some ball contained in it, and hence our function is valid (it never maps to the empty set). Now starting at a point in the interior of , DC gives us the function all whose elements are constrained by acting on the previous value. (This is all proven in this lemma.) Now is a sequence of tagged open balls, forming an inclusion chain (see bcthlem2 19231) and whose sizes tend to zero, since they are bounded above by . Thus, the centers of these balls form a Cauchy sequence, and converge to a point (see bcthlem4 19233). Since the inclusion chain also ensures the closure of each ball is in the previous ball, the point must be in all these balls (see bcthlem3 19232) and hence misses each , contradicting the fact that is in the interior of (which was the starting point). (Contributed by Mario Carneiro, 6-Jan-2014.)

Theorembcth 19235* Baire's Category Theorem. If a nonempty metric space is complete, it is nonmeager in itself. In other words, no open set in the metric space can be the countable union of rare closed subsets (where rare means having an empty interior), so some subset must have a nonempty interior. Theorem 4.7-2 of [Kreyszig] p. 247. (The terminology "meager" and "nonmeager" is used by Kreyszig to replace Baire's "of the first category" and "of the second category." The latter terms are going out of favor to avoid confusion with category theory.) See bcthlem5 19234 for an overview of the proof. (Contributed by NM, 28-Oct-2007.) (Proof shortened by Mario Carneiro, 6-Jan-2014.)

Theorembcth2 19236* Baire's Category Theorem, version 2: If countably many closed sets cover , then one of them has an interior. (Contributed by Mario Carneiro, 10-Jan-2014.)

Theorembcth3 19237* Baire's Category Theorem, version 3: The intersection of countably many dense open sets is dense. (Contributed by Mario Carneiro, 10-Jan-2014.)

11.5.5  Banach spaces and complex Hilbert spaces

Syntaxccms 19238 Extend class notation with the class of all complete normed groups.
CMetSp

Syntaxcbn 19239 Extend class notation with the class of all Banach spaces.
Ban

Syntaxchl 19240 Extend class notation with the class of all complex Hilbert spaces.

Definitiondf-cms 19241* Define the class of all complete metric spaces. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp

Definitiondf-bn 19242 Define the class of all Banach spaces. A Banach space is a normed vector space such that both the vector space and the scalar field are complete under their respective norm-induced metrics. (Contributed by NM, 5-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2015.)
Ban NrmVec CMetSp Scalar CMetSp

Definitiondf-hl 19243 Define the class of all complex Hilbert spaces. A complex Hilbert space is a Banach space which is also an inner product space over the complex numbers. (Contributed by Steve Rodriguez, 28-Apr-2007.)
Ban

Theoremisbn 19244 A Banach space is a normed vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2015.)
Scalar       Ban NrmVec CMetSp CMetSp

Theorembnsca 19245 The scalar field of a complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
Scalar       Ban CMetSp

Theorembnnvc 19246 A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Ban NrmVec

Theorembnnlm 19247 A Banach space is a normed module. (Contributed by Mario Carneiro, 15-Oct-2015.)
Ban NrmMod

Theorembnngp 19248 A Banach space is a normed group. (Contributed by Mario Carneiro, 15-Oct-2015.)
Ban NrmGrp

Theorembnlmod 19249 A Banach space is a left module. (Contributed by Mario Carneiro, 15-Oct-2015.)
Ban

Theorembncms 19250 A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Ban CMetSp

Theoremiscms 19251 A complete metric space is a metric space with a complete metric. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp

Theoremcmscmet 19252 The induced metric on a complete normed group is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp

Theorembncmet 19253 The induced metric on Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
Ban

Theoremcmsms 19254 A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp

Theoremcmspropd 19255 Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp CMetSp

Theoremcmsss 19256 The restriction of a complete metric space is complete iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
s                      CMetSp CMetSp

Theoremlssbn 19257 A subspace of a Banach space is a Banach space iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
s                      Ban Ban

Theoremcmetcusp1OLD 19258 If the uniform set of a complete metric space is the uniform structure generated by its metric, then it is a complete uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
UnifSt       CMetSp metUnifOLD CUnifSp

Theoremcmetcusp1 19259 If the uniform set of a complete metric space is the uniform structure generated by its metric, then it is a complete uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
UnifSt       CMetSp metUnif CUnifSp

TheoremcmetcuspOLD 19260 The uniform space generated by a complete metric is a complete uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSpmetUnifOLD CUnifSp

Theoremcmetcusp 19261 The uniform space generated by a complete metric is a complete uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSpmetUnif CUnifSp

Theoremcncms 19262 The field of complex numbers is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
fld CMetSp

Theoremcnflduss 19263 The uniform structure of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
UnifStfld       metUnif

Theoremcnfldcusp 19264 The field of complex numbers is a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
fld CUnifSp

Theoremresscdrg 19265 The real numbers are a subset of any complete subfield in the complexes. (Contributed by Mario Carneiro, 15-Oct-2015.)
flds        SubRingfld CMetSp

Theoremcncdrg 19266 The only complete subfields of the complexes are and . (Contributed by Mario Carneiro, 15-Oct-2015.)
flds        SubRingfld CMetSp

Theoremsrabn 19267 The subring algebra over a complete normed ring is a Banach space iff the subring is a closed division ring. (Contributed by Mario Carneiro, 15-Oct-2015.)
subringAlg               NrmRing CMetSp SubRing Ban s

Theoremrlmbn 19268 The ring module over a complete normed division ring is a Banach space. (Contributed by Mario Carneiro, 15-Oct-2015.)
NrmRing CMetSp ringLMod Ban

Theoremishl 19269 The predicate "is a complex Hilbert space." A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
Ban

Theoremhlbn 19270 Every complex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
Ban

Theoremhlcph 19271 Every complex Hilbert space is a complex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremhlphl 19272 Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)

Theoremhlcms 19273 Every complex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.)
CMetSp

Theoremhlprlem 19274 Lemma for hlpr 19276. (Contributed by Mario Carneiro, 15-Oct-2015.)
Scalar              SubRingfld flds flds CMetSp

Theoremhlress 19275 The scalar field of a complex Hilbert space contains . (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar

Theoremhlpr 19276 The scalar field of a complex Hilbert space is either or . (Contributed by Mario Carneiro, 15-Oct-2015.)
Scalar

Theoremishl2 19277 A Hilbert space is a complete complex pre-Hilbert space over or . (Contributed by Mario Carneiro, 15-Oct-2015.)
Scalar              CMetSp

11.5.6  Minimizing Vector Theorem

Theoremminveclem1 19278* Lemma for minvec 19290. The set of all distances from points of to are a nonempty set of nonnegative reals. (Contributed by Mario Carneiro, 8-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem4c 19279* Lemma for minvec 19290. The infimum of the distances to is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem2 19280* Lemma for minvec 19290. Any two points and in are close to each other if they are close to the infimum of distance to . (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem3a 19281* Lemma for minvec 19290. is a complete metric when restricted to . (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem3b 19282* Lemma for minvec 19290. The set of vectors within a fixed distance of the infimum forms a filter base. (Contributed by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem3 19283* Lemma for minvec 19290. The filter formed by taking elements successively closer to the infimum is Cauchy. (Contributed by Mario Carneiro, 8-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp                                                 CauFil

Theoremminveclem4a 19284* Lemma for minvec 19290. converges to a point in . (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem4b 19285* Lemma for minvec 19290. The convergent point of the Cauchy sequence is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem4 19286* Lemma for minvec 19290. The convergent point of the Cauchy sequence attains the minimum distance, and so is closer to than any other point in . (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem5 19287* Lemma for minvec 19290. Discharge the assumptions in minveclem4 19286. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem6 19288* Lemma for minvec 19290. Any minimal point is less than away from . (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminveclem7 19289* Lemma for minvec 19290. Since any two minimal points are distance zero away from each other, the minimal point is unique. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

Theoremminvec 19290* Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace that minimizes the distance to an arbitrary vector in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.)
s CMetSp

11.5.7  Projection Theorem

Theorempjthlem1 19291* Lemma for pjth 19293. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 17-Oct-2015.)

Theorempjthlem2 19292 Lemma for pjth 19293. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.)

Theorempjth 19293 Projection Theorem: Any Hilbert space vector can be decomposed uniquely into a member of a closed subspace and a member of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.)

Theorempjth2 19294 Projection Theorem with abbreviations: A topologically closed subspace is a projection subspace. (Contributed by Mario Carneiro, 17-Oct-2015.)

Theoremcldcss 19295 Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)

Theoremcldcss2 19296 Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.)

Theoremhlhil 19297 Corollary of the Projection Theorem: A complex Hilbert space is a Hilbert space (in the algebraic sense, meaning that all algebraically closed subspaces have a projection decomposition). (Contributed by Mario Carneiro, 17-Oct-2015.)

PART 12  BASIC REAL AND COMPLEX ANALYSIS

12.1  Continuity

12.1.1  Intermediate value theorem

Theorempmltpclem1 19298* Lemma for pmltpc 19300. (Contributed by Mario Carneiro, 1-Jul-2014.)

Theorempmltpclem2 19299* Lemma for pmltpc 19300. (Contributed by Mario Carneiro, 1-Jul-2014.)

Theorempmltpc 19300* Any function on the reals is either increasing, decreasing, or has a triple of points in a vee formation. (This theorem was created on demand by Mario Carneiro for the 6PCM conference in Bialystok, 1-Jul-2014.) (Contributed by Mario Carneiro, 1-Jul-2014.)

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