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

Theoremnmparlem 19201 Lemma for nmpar 19202. (Contributed by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremnmpar 19202 A complex pre-Hilbert space satisfies the parallelogram law. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremipcnlem2 19203 The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremipcnlem1 19204* The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremipcn 19205 The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)
fld

Theoremcnmpt1ip 19206* Continuity of inner product; analogue of cnmpt12f 17703 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.)
fld                     TopOn

Theoremcnmpt2ip 19207* Continuity of inner product; analogue of cnmpt22f 17712 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.)
fld                     TopOn       TopOn

Theoremcsscld 19208 A "closed subspace" in a complex pre-Hilbert space is actually closed in the topology induced by the norm, thus justifying the terminology "closed subspace". (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremclsocv 19209 The orthogonal complement of the closure of a subset is the same as the orthogonal complement of the subset itself. (Contributed by Mario Carneiro, 13-Oct-2015.)

11.5.3  Convergence and completeness

Syntaxccfil 19210 Extend class notation with the set of Cauchy filters.
CauFil

Syntaxcca 19211 Extend class notation with a function on metric spaces whose value is the set of all Cauchy sequences of the space.

Syntaxcms 19212 Extend class notation with class of complete metric spaces.

Definitiondf-cfil 19213* Define the set of Cauchy filters on a metric space. A Cauchy filter is a filter on the set such that for every there is an element of the filter whose metric diameter is less than . (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Definitiondf-cau 19214* Define a function on metric spaces whose value is the set of Cauchy sequences of the space. (Contributed by NM, 8-Sep-2006.)

Definitiondf-cmet 19215* Define the class of complete metrics. (Contributed by Mario Carneiro, 1-May-2014.)
CauFil

Theoremlmmbr 19216* Express the binary relation "sequence converges to point " in a metric space. Definition 1.4-1 of [Kreyszig] p. 25. The condition allows us to use objects more general than sequences when convenient; see the comment in df-lm 17298. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremlmmbr2 19217* Express the binary relation "sequence converges to point " in a metric space. Definition 1.4-1 of [Kreyszig] p. 25. The condition allows us to use objects more general than sequences when convenient; see the comment in df-lm 17298. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremlmmbr3 19218* Express the binary relation "sequence converges to point " in a metric space using an abitrary set of upper integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremlmmcvg 19219* Convergence property of a converging sequence. (Contributed by NM, 1-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremlmmbrf 19220* Express the binary relation "sequence converges to point " in a metric space using an abitrary set of upper integers. This version of lmmbr2 19217 presupposes that is a function. (Contributed by NM, 20-Jul-2007.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremlmnn 19221* A condition that implies convergence. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremcfilfval 19222* The set of Cauchy filters on a metric space. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremiscfil 19223* The property of being a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremiscfil2 19224* The property of being a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremcfilfil 19225 A Cauchy filter is a filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremcfili 19226* Property of a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremcfil3i 19227* A Cauchy filter contains balls of any pre-chosen size. (Contributed by Mario Carneiro, 15-Oct-2015.)
CauFil

Theoremcfilss 19228 A filter finer than a Cauchy filter is Cauchy. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil CauFil

Theoremfgcfil 19229* The Cauchy filter condition for a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremfmcfil 19230* The Cauchy filter condition for a filter map. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremiscfil3 19231* A filter is Cauchy iff it contains a ball of any chosen size. (Contributed by Mario Carneiro, 15-Oct-2015.)
CauFil

Theoremcfilfcls 19232 Similar to ultrafilters (uffclsflim 18068), the cluster points and limit points of a Cauchy filter coincide. (Contributed by Mario Carneiro, 15-Oct-2015.)
CauFil

Theoremcaufval 19233* The set of Cauchy sequences on a metric space. (Contributed by NM, 8-Sep-2006.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremiscau 19234* Express the property " is a Cauchy sequence of metric ." Part of Definition 1.4-3 of [Kreyszig] p. 28. The condition allows us to use objects more general than sequences when convenient; see the comment in df-lm 17298. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

Theoremiscau2 19235* Express the property " is a Cauchy sequence of metric ," using an abitrary set of upper integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

Theoremiscau3 19236* Express the Cauchy sequence property in the more conventional three-quantifier form. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

Theoremiscau4 19237* Express the property " is a Cauchy sequence of metric ," using an arbitrary set of upper integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)

Theoremiscauf 19238* Express the property " is a Cauchy sequence of metric " presupposing is a function. (Contributed by NM, 24-Jul-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)

Theoremcaun0 19239 A metric with a Cauchy sequence cannot be empty. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)

Theoremcaufpm 19240 Inclusion of a Cauchy sequence, under our definition. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)

Theoremcaucfil 19241 A Cauchy sequence predicate can be expressed in terms of the Cauchy filter predicate for a suitably chosen filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremiscmet 19242* The property " is a complete metric." meaning all Cauchy filters converge to a point in the space. (Contributed by Mario Carneiro, 1-May-2014.) (Revised by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremcmetcvg 19243 The convergence of a Cauchy filter in a complete metric space. (Contributed by Mario Carneiro, 14-Oct-2015.)
CauFil

Theoremcmetmet 19244 A complete metric space is a metric space. (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 29-Jan-2014.)

Theoremcmetmeti 19245 A complete metric space is a metric space. (Contributed by NM, 26-Oct-2007.)

Theoremcmetcaulem 19246* Lemma for cmetcau 19247. (Contributed by Mario Carneiro, 14-Oct-2015.)

Theoremcmetcau 19247 The convergence of a Cauchy sequence in a complete metric space. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Oct-2015.)

Theoremiscmet3lem3 19248* Lemma for iscmet3 19251. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremiscmet3lem1 19249* Lemma for iscmet3 19251. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremiscmet3lem2 19250* Lemma for iscmet3 19251. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremiscmet3 19251* The property " is a complete metric" expressed in terms of functions on (or any other upper integer set). Thus, we only have to look at functions on , and not all possible Cauchy filters, to determine completeness. (The proof uses countable choice.) (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 5-May-2014.)

Theoremiscmet2 19252 A metric is complete iff all Cauchy sequences converge to a point in the space. The proof uses countable choice. Part of Definition 1.4-3 of [Kreyszig] p. 28. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 15-Oct-2015.)

Theoremcfilresi 19253 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 19254 Cauchy filter on a metric subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
CauFil t CauFil

Theoremcaussi 19255 Cauchy sequence on a metric subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)

Theoremcauss 19256 Cauchy sequence on a metric subspace. (Contributed by NM, 29-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)

Theoremequivcfil 19257* 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 19258* 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 19259* 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 19260 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 19261 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 19262* 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 8320. 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 19263* 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 19264 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 19265* 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 19266* 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 19267* 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 19268* 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 19269* Properties that determine a complete metric space. (Contributed by NM, 15-Apr-2007.) (Revised by Mario Carneiro, 5-May-2014.)

Theoremlmcau 19270 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 19271 Every convergent filter in a metric space is a Cauchy filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
CauFil

Theoremcmetss 19272 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 19273* If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 19258, metss2 18547, 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 19274* 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 19275 A compact metric space is complete. One half of heibor 26543. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremcfilucfil3OLD 19276 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 19277 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 19278 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 19279 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 19280 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 19281 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 19282* Lemma for bcth 19287. Substitutions for the function . (Contributed by Mario Carneiro, 9-Jan-2014.)

Theorembcthlem2 19283* Lemma for bcth 19287. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)

Theorembcthlem3 19284* Lemma for bcth 19287. 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 19285* Lemma for bcth 19287. 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 19286* Lemma for bcth 19287. The proof makes essential use of the Axiom of Dependent Choice axdc4uz 11327, 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 19283) 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 19285). 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 19284) 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 19287* 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 19286 for an overview of the proof. (Contributed by NM, 28-Oct-2007.) (Proof shortened by Mario Carneiro, 6-Jan-2014.)

Theorembcth2 19288* 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 19289* 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 19290 Extend class notation with the class of all complete normed groups.
CMetSp

Syntaxcbn 19291 Extend class notation with the class of all Banach spaces.
Ban

Syntaxchl 19292 Extend class notation with the class of all complex Hilbert spaces.

Definitiondf-cms 19293* Define the class of all complete metric spaces. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp

Definitiondf-bn 19294 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 19295 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 19296 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 19297 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 19298 A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Ban NrmVec

Theorembnnlm 19299 A Banach space is a normed module. (Contributed by Mario Carneiro, 15-Oct-2015.)
Ban NrmMod

Theorembnngp 19300 A Banach space is a normed group. (Contributed by Mario Carneiro, 15-Oct-2015.)
Ban NrmGrp

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