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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lmclim 23101 | 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.) |
ℂfld | ||
Theorem | lmclimf 23102 | 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.) |
ℂfld | ||
Theorem | metelcls 23103* | 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 9257. 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.) |
Theorem | metcld 23104* | 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.) |
Theorem | metcld2 23105 | 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.) |
Theorem | caubl 23106* | 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.) |
Theorem | caublcls 23107* | 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.) |
Theorem | metcnp4 23108* | 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.) |
Theorem | metcn4 23109* | 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.) |
Theorem | iscmet3i 23110* | Properties that determine a complete metric space. (Contributed by NM, 15-Apr-2007.) (Revised by Mario Carneiro, 5-May-2014.) |
Theorem | lmcau 23111 | 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.) |
Theorem | flimcfil 23112 | Every convergent filter in a metric space is a Cauchy filter. (Contributed by Mario Carneiro, 15-Oct-2015.) |
CauFil | ||
Theorem | cmetss 23113 | 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.) |
Theorem | equivcmet 23114* | If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 23098, metss2 22317, 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.) |
Theorem | relcmpcmet 23115* | 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 | ||
Theorem | cmpcmet 23116 | A compact metric space is complete. One half of heibor 33620. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Theorem | cfilucfil3 23117 | 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 | ||
Theorem | cfilucfil4 23118 | 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 | ||
Theorem | cncmet 23119 | 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.) |
Theorem | recmet 23120 | The real numbers are a complete metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
Theorem | bcthlem1 23121* | Lemma for bcth 23126. Substitutions for the function . (Contributed by Mario Carneiro, 9-Jan-2014.) |
Theorem | bcthlem2 23122* | Lemma for bcth 23126. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.) |
Theorem | bcthlem3 23123* | Lemma for bcth 23126. 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.) |
Theorem | bcthlem4 23124* | Lemma for bcth 23126. 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.) |
Theorem | bcthlem5 23125* |
Lemma for bcth 23126. The proof makes essential use of the Axiom
of
Dependent Choice axdc4uz 12783, 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 23122) 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 23124). 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 23123) 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.) |
Theorem | bcth 23126* | 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 a closure with 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 23125 for an overview of the proof. (Contributed by NM, 28-Oct-2007.) (Proof shortened by Mario Carneiro, 6-Jan-2014.) |
Theorem | bcth2 23127* | 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.) |
Theorem | bcth3 23128* | Baire's Category Theorem, version 3: The intersection of countably many dense open sets is dense. (Contributed by Mario Carneiro, 10-Jan-2014.) |
Syntax | ccms 23129 | Extend class notation with the class of all complete normed groups. |
CMetSp | ||
Syntax | cbn 23130 | Extend class notation with the class of all Banach spaces. |
Ban | ||
Syntax | chl 23131 | Extend class notation with the class of all subcomplex Hilbert spaces. |
Definition | df-cms 23132* | Define the class of all complete metric spaces. (Contributed by Mario Carneiro, 15-Oct-2015.) |
CMetSp | ||
Definition | df-bn 23133 | 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 | ||
Definition | df-hl 23134 | Define the class of all subcomplex Hilbert spaces. A subcomplex Hilbert space is a Banach space which is also an inner product space over a quadratically closed subfield of the field of complex numbers. (Contributed by Steve Rodriguez, 28-Apr-2007.) |
Ban | ||
Theorem | isbn 23135 | 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 | ||
Theorem | bnsca 23136 | The scalar field of a Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.) |
Scalar Ban CMetSp | ||
Theorem | bnnvc 23137 | A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ban NrmVec | ||
Theorem | bnnlm 23138 | A Banach space is a normed module. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ban NrmMod | ||
Theorem | bnngp 23139 | A Banach space is a normed group. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ban NrmGrp | ||
Theorem | bnlmod 23140 | A Banach space is a left module. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ban | ||
Theorem | bncms 23141 | A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ban CMetSp | ||
Theorem | iscms 23142 | A complete metric space is a metric space with a complete metric. (Contributed by Mario Carneiro, 15-Oct-2015.) |
CMetSp | ||
Theorem | cmscmet 23143 | The induced metric on a complete normed group is complete. (Contributed by Mario Carneiro, 15-Oct-2015.) |
CMetSp | ||
Theorem | bncmet 23144 | The induced metric on Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.) |
Ban | ||
Theorem | cmsms 23145 | A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
CMetSp | ||
Theorem | cmspropd 23146 | Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
CMetSp CMetSp | ||
Theorem | cmsss 23147 | The restriction of a complete metric space is complete iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.) |
↾s CMetSp CMetSp | ||
Theorem | lssbn 23148 | A subspace of a Banach space is a Banach space iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.) |
↾s Ban Ban | ||
Theorem | cmetcusp1 23149 | 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 | ||
Theorem | cmetcusp 23150 | The uniform space generated by a complete metric is a complete uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.) |
toUnifSpmetUnif CUnifSp | ||
Theorem | cncms 23151 | The field of complex numbers is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
ℂfld CMetSp | ||
Theorem | cnflduss 23152 | The uniform structure of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
UnifStℂfld metUnif | ||
Theorem | cnfldcusp 23153 | The field of complex numbers is a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.) |
ℂfld CUnifSp | ||
Theorem | resscdrg 23154 | The real numbers are a subset of any complete subfield in the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.) |
ℂfld ↾s SubRingℂfld CMetSp | ||
Theorem | cncdrg 23155 | The only complete subfields of the complex numbers are and . (Contributed by Mario Carneiro, 15-Oct-2015.) |
ℂfld ↾s SubRingℂfld CMetSp | ||
Theorem | srabn 23156 | 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 | ||
Theorem | rlmbn 23157 | The ring module over a complete normed division ring is a Banach space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
NrmRing CMetSp ringLMod Ban | ||
Theorem | ishl 23158 | The predicate "is a subcomplex 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 | ||
Theorem | hlbn 23159 | Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) |
Ban | ||
Theorem | hlcph 23160 | Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Theorem | hlphl 23161 | Every subcomplex 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.) |
Theorem | hlcms 23162 | Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.) |
CMetSp | ||
Theorem | hlprlem 23163 | Lemma for hlpr 23165. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Scalar SubRingℂfld ℂfld ↾s ℂfld ↾s CMetSp | ||
Theorem | hlress 23164 | The scalar field of a subcomplex Hilbert space contains . (Contributed by Mario Carneiro, 8-Oct-2015.) |
Scalar | ||
Theorem | hlpr 23165 | The scalar field of a subcomplex Hilbert space is either or . (Contributed by Mario Carneiro, 15-Oct-2015.) |
Scalar | ||
Theorem | ishl2 23166 | A Hilbert space is a complete subcomplex pre-Hilbert space over or . (Contributed by Mario Carneiro, 15-Oct-2015.) |
Scalar CMetSp | ||
Theorem | retopn 23167 | The topology of the real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
RRfld | ||
Theorem | recms 23168 | The real numbers form a complete metric space. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
RRfld CMetSp | ||
Theorem | reust 23169 | The Uniform structure of the real numbers. (Contributed by Thierry Arnoux, 14-Feb-2018.) |
UnifStRRfld metUnifRRfld | ||
Theorem | recusp 23170 | The real numbers form a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.) |
RRfld CUnifSp | ||
Syntax | crrx 23171 | Extend class notation with generalized real Euclidean spaces. |
ℝ^ | ||
Syntax | cehl 23172 | Extend class notation with real Euclidean spaces. |
𝔼hil | ||
Definition | df-rrx 23173 | Define the function associating with a set the free real vector space on that set, equipped with the natural inner product. This is the direct sum of copies of the field of real numbers indexed by that set. We call it here a "generalized real Euclidean space", but note that it need not be complete (for instance if the given set is infinite countable). (Contributed by Thierry Arnoux, 16-Jun-2019.) |
ℝ^ toCHilRRfld freeLMod | ||
Definition | df-ehl 23174 | Define a function generating the real Euclidean spaces of finite dimension. The case corresponds to a space of dimension 0, that is, limited to a neutral element. Members of this family of spaces are Hilbert spaces, as shown in - ehlhl . (Contributed by Thierry Arnoux, 16-Jun-2019.) |
𝔼hil ℝ^ | ||
Theorem | rrxval 23175 | Value of the generalized Euclidean space. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
ℝ^ toCHilRRfld freeLMod | ||
Theorem | rrxbase 23176* | The base of the generalized real Euclidean space is the set of functions with finite support. (Contributed by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 22-Jul-2019.) |
ℝ^ finSupp | ||
Theorem | rrxprds 23177 | Expand the definition of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
ℝ^ toCHilRRflds subringAlg RRfld ↾s | ||
Theorem | rrxip 23178* | The inner product of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
ℝ^ RRfld g | ||
Theorem | rrxnm 23179* | The norm of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
ℝ^ RRfld g | ||
Theorem | rrxcph 23180 | Generalized Euclidean real spaces are pre-Hilbert spaces. (Contributed by Thierry Arnoux, 23-Jun-2019.) (Proof shortened by AV, 22-Jul-2019.) |
ℝ^ | ||
Theorem | rrxds 23181* | The distance over generalized Euclidean spaces. Compare with df-rrn 33625. (Contributed by Thierry Arnoux, 20-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.) |
ℝ^ RRfld g | ||
Theorem | csbren 23182* | Cauchy-Schwarz-Bunjakovsky inequality for R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.) |
Theorem | trirn 23183* | Triangle inequality in R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.) |
Theorem | rrxf 23184* | Euclidean vectors as functions. (Contributed by Thierry Arnoux, 7-Jul-2019.) |
finSupp | ||
Theorem | rrxfsupp 23185* | Euclidean vectors are of finite support. (Contributed by Thierry Arnoux, 7-Jul-2019.) |
finSupp supp | ||
Theorem | rrxsuppss 23186* | Support of Euclidean vectors. (Contributed by Thierry Arnoux, 7-Jul-2019.) |
finSupp supp | ||
Theorem | rrxmvallem 23187* | Support of the function used for building the distance . (Contributed by Thierry Arnoux, 30-Jun-2019.) |
finSupp supp supp supp | ||
Theorem | rrxmval 23188* | The value of the Euclidean metric. Compare with rrnmval 33627. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
finSupp ℝ^ supp supp | ||
Theorem | rrxmfval 23189* | The value of the Euclidean metric. Compare with rrnval 33626. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
finSupp ℝ^ supp supp | ||
Theorem | rrxmetlem 23190* | Lemma for rrxmet 23191. (Contributed by Thierry Arnoux, 5-Jul-2019.) |
finSupp ℝ^ supp supp supp supp | ||
Theorem | rrxmet 23191* | Euclidean space is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.) (Revised by Thierry Arnoux, 30-Jun-2019.) |
finSupp ℝ^ | ||
Theorem | rrxdstprj1 23192* | The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 7-Jul-2019.) |
finSupp ℝ^ | ||
Theorem | ehlval 23193 | Value of the Euclidean space of dimension . (Contributed by Thierry Arnoux, 16-Jun-2019.) |
𝔼hil ℝ^ | ||
Theorem | ehlbase 23194 | The base of the Euclidean space is the set of n-tuples of real numbers. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
𝔼hil | ||
Theorem | minveclem1 23195* | Lemma for minvec 23207. 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 | ||
Theorem | minveclem4c 23196* | Lemma for minvec 23207. The infimum of the distances to is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.) |
↾s CMetSp inf | ||
Theorem | minveclem2 23197* | Lemma for minvec 23207. 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.) (Revised by AV, 3-Oct-2020.) |
↾s CMetSp inf | ||
Theorem | minveclem3a 23198* | Lemma for minvec 23207. is a complete metric when restricted to . (Contributed by Mario Carneiro, 7-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) |
↾s CMetSp inf | ||
Theorem | minveclem3b 23199* | Lemma for minvec 23207. The set of vectors within a fixed distance of the infimum forms a filter base. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 3-Oct-2020.) |
↾s CMetSp inf | ||
Theorem | minveclem3 23200* | Lemma for minvec 23207. 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 inf CauFil |
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