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

Theoremlmmbr2 18701* 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 16975. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremlmmbr3 18702* 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 18703* Convergence property of a converging sequence. (Contributed by NM, 1-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremlmmbrf 18704* Express the binary relation "sequence converges to point " in a metric space using an abitrary set of upper integers. This version of lmmbr2 18701 presupposes that is a function. (Contributed by NM, 20-Jul-2007.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremlmnn 18705* A condition that implies convergence. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremcfilfval 18706* The set of Cauchy filters on a metric space. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremiscfil 18707* The property of being a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremiscfil2 18708* The property of being a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremcfilfil 18709 A Cauchy filter is a filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremcfili 18710* Property of a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremcfil3i 18711* A Cauchy filter contains balls of any pre-chosen size. (Contributed by Mario Carneiro, 15-Oct-2015.)
CauFil

Theoremcfilss 18712 A filter finer than a Cauchy filter is Cauchy. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil CauFil

Theoremfgcfil 18713* The Cauchy filter condition for a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremfmcfil 18714* The Cauchy filter condition for a filter map. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Theoremiscfil3 18715* A filter is Cauchy iff it contains a ball of any chosen size. (Contributed by Mario Carneiro, 15-Oct-2015.)
CauFil

Theoremcfilfcls 18716 Similar to ultrafilters (uffclsflim 17742), the cluster points and limit points of a Cauchy filter coincide. (Contributed by Mario Carneiro, 15-Oct-2015.)
CauFil

Theoremcaufval 18717* The set of Cauchy sequences on a metric space. (Contributed by NM, 8-Sep-2006.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremiscau 18718* 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 16975. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

Theoremiscau2 18719* 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 18720* 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 18721* 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 18722* 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 18723 A metric with a Cauchy sequence cannot be empty. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)

Theoremcaufpm 18724 Inclusion of a Cauchy sequence, under our definition. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)

Theoremcaucfil 18725 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 18726* 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 18727 The convergence of a Cauchy filter in a complete metric space. (Contributed by Mario Carneiro, 14-Oct-2015.)
CauFil

Theoremcmetmet 18728 A complete metric space is a metric space. (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 29-Jan-2014.)

Theoremcmetmeti 18729 A complete metric space is a metric space. (Contributed by NM, 26-Oct-2007.)

Theoremcmetcaulem 18730* Lemma for cmetcau 18731. (Contributed by Mario Carneiro, 14-Oct-2015.)

Theoremcmetcau 18731 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 18732* Lemma for iscmet3 18735. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremiscmet3lem1 18733* Lemma for iscmet3 18735. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremiscmet3lem2 18734* Lemma for iscmet3 18735. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremiscmet3 18735* 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 18736 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 18737 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 18738 Cauchy filter on a metric subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
CauFil t CauFil

Theoremcaussi 18739 Cauchy sequence on a metric subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)

Theoremcauss 18740 Cauchy sequence on a metric subspace. (Contributed by NM, 29-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)

Theoremequivcfil 18741* 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 18742* 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 18743* 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 18744 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 18745 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 18746* 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 8077. 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 18747* 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 18748 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 18749* 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 18750* 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 18751* 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 18752* 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 18753* Properties that determine a complete metric space. (Contributed by NM, 15-Apr-2007.) (Revised by Mario Carneiro, 5-May-2014.)

Theoremlmcau 18754 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 18755 Every convergent filter in a metric space is a Cauchy filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
CauFil

Theoremcmetss 18756 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 18757* If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 18742, metss2 18074, 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 18758* 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 18759 A compact metric space is complete. One half of heibor 26648. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremcncmet 18760 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 18761 The real numbers are a complete metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

11.4.4  Baire's Category Theorem

Theorembcthlem1 18762* Lemma for bcth 18767. Substitutions for the function . (Contributed by Mario Carneiro, 9-Jan-2014.)

Theorembcthlem2 18763* Lemma for bcth 18767. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)

Theorembcthlem3 18764* Lemma for bcth 18767. 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 18765* Lemma for bcth 18767. 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 18766* Lemma for bcth 18767. The proof makes essential use of the Axiom of Dependent Choice axdc4uz 11061, 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 18763) 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 18765). 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 18764) 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 18767* 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 18766 for an overview of the proof. (Contributed by NM, 28-Oct-2007.) (Proof shortened by Mario Carneiro, 6-Jan-2014.)

Theorembcth2 18768* 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 18769* Baire's Category Theorem, version 3: The intersection of countably many dense open sets is dense. (Contributed by Mario Carneiro, 10-Jan-2014.)

11.4.5  Banach spaces and complex Hilbert spaces

Syntaxccms 18770 Extend class notation with the class of all complete normed groups.
CMetSp

Syntaxcbn 18771 Extend class notation with the class of all Banach spaces.
Ban

Syntaxchl 18772 Extend class notation with the class of all complex Hilbert spaces.

Definitiondf-cms 18773* Define the class of all complete metric spaces. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp

Definitiondf-bn 18774 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 18775 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 18776 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 18777 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 18778 A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Ban NrmVec

Theorembnnlm 18779 A Banach space is a normed module. (Contributed by Mario Carneiro, 15-Oct-2015.)
Ban NrmMod

Theorembnngp 18780 A Banach space is a normed group. (Contributed by Mario Carneiro, 15-Oct-2015.)
Ban NrmGrp

Theorembnlmod 18781 A Banach space is a left module. (Contributed by Mario Carneiro, 15-Oct-2015.)
Ban

Theorembncms 18782 A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Ban CMetSp

Theoremiscms 18783 A complete metric space is a metric space with a complete metric. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp

Theoremcmscmet 18784 The induced metric on a complete normed group is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp

Theorembncmet 18785 The induced metric on Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
Ban

Theoremcmsms 18786 A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp

Theoremcmspropd 18787 Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp CMetSp

Theoremcmsss 18788 The restriction of a complete metric space is complete iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
s                      CMetSp CMetSp

Theoremlssbn 18789 A subspace of a Banach space is a Banach space iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
s                      Ban Ban

Theoremcncms 18790 The field of complex numbers is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
fld CMetSp

Theoremresscdrg 18791 The real numbers are a subset of any complete subfield in the complexes. (Contributed by Mario Carneiro, 15-Oct-2015.)
flds        SubRingfld CMetSp

Theoremcncdrg 18792 The only complete subfields of the complexes are and . (Contributed by Mario Carneiro, 15-Oct-2015.)
flds        SubRingfld CMetSp

Theoremsrabn 18793 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 18794 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 18795 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 18796 Every complex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
Ban

Theoremhlcph 18797 Every complex Hilbert space is a complex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremhlphl 18798 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 18799 Every complex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.)
CMetSp

Theoremhlprlem 18800 Lemma for hlpr 18802. (Contributed by Mario Carneiro, 15-Oct-2015.)
Scalar              SubRingfld flds flds CMetSp

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