![]() |
Metamath
Proof Explorer Theorem List (p. 249 of 480) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30209) |
![]() (30210-31732) |
![]() (31733-47936) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | iscau2 24801* | Express the property "πΉ is a Cauchy sequence of metric π· " using an arbitrary upper set of integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.) |
β’ (π· β (βMetβπ) β (πΉ β (Cauβπ·) β (πΉ β (π βpm β) β§ βπ₯ β β+ βπ β β€ βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π β§ ((πΉβπ)π·(πΉβπ)) < π₯)))) | ||
Theorem | iscau3 24802* | 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.) |
β’ π = (β€β₯βπ) & β’ (π β π· β (βMetβπ)) & β’ (π β π β β€) β β’ (π β (πΉ β (Cauβπ·) β (πΉ β (π βpm β) β§ βπ₯ β β+ βπ β π βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π β§ βπ β (β€β₯βπ)((πΉβπ)π·(πΉβπ)) < π₯)))) | ||
Theorem | iscau4 24803* | Express the property "πΉ is a Cauchy sequence of metric π· " using an arbitrary upper set of integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.) |
β’ π = (β€β₯βπ) & β’ (π β π· β (βMetβπ)) & β’ (π β π β β€) & β’ ((π β§ π β π) β (πΉβπ) = π΄) & β’ ((π β§ π β π) β (πΉβπ) = π΅) β β’ (π β (πΉ β (Cauβπ·) β (πΉ β (π βpm β) β§ βπ₯ β β+ βπ β π βπ β (β€β₯βπ)(π β dom πΉ β§ π΄ β π β§ (π΄π·π΅) < π₯)))) | ||
Theorem | iscauf 24804* | Express the property "πΉ is a Cauchy sequence of metric π· " presupposing πΉ is a function. (Contributed by NM, 24-Jul-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) |
β’ π = (β€β₯βπ) & β’ (π β π· β (βMetβπ)) & β’ (π β π β β€) & β’ ((π β§ π β π) β (πΉβπ) = π΄) & β’ ((π β§ π β π) β (πΉβπ) = π΅) & β’ (π β πΉ:πβΆπ) β β’ (π β (πΉ β (Cauβπ·) β βπ₯ β β+ βπ β π βπ β (β€β₯βπ)(π΅π·π΄) < π₯)) | ||
Theorem | caun0 24805 | A metric with a Cauchy sequence cannot be empty. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.) |
β’ ((π· β (βMetβπ) β§ πΉ β (Cauβπ·)) β π β β ) | ||
Theorem | caufpm 24806 | Inclusion of a Cauchy sequence, under our definition. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.) |
β’ ((π· β (βMetβπ) β§ πΉ β (Cauβπ·)) β πΉ β (π βpm β)) | ||
Theorem | caucfil 24807 | 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.) |
β’ π = (β€β₯βπ) & β’ πΏ = ((π FilMap πΉ)β(β€β₯ β π)) β β’ ((π· β (βMetβπ) β§ π β β€ β§ πΉ:πβΆπ) β (πΉ β (Cauβπ·) β πΏ β (CauFilβπ·))) | ||
Theorem | iscmet 24808* | The property "π· 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.) |
β’ π½ = (MetOpenβπ·) β β’ (π· β (CMetβπ) β (π· β (Metβπ) β§ βπ β (CauFilβπ·)(π½ fLim π) β β )) | ||
Theorem | cmetcvg 24809 | The convergence of a Cauchy filter in a complete metric space. (Contributed by Mario Carneiro, 14-Oct-2015.) |
β’ π½ = (MetOpenβπ·) β β’ ((π· β (CMetβπ) β§ πΉ β (CauFilβπ·)) β (π½ fLim πΉ) β β ) | ||
Theorem | cmetmet 24810 | A complete metric space is a metric space. (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 29-Jan-2014.) |
β’ (π· β (CMetβπ) β π· β (Metβπ)) | ||
Theorem | cmetmeti 24811 | A complete metric space is a metric space. (Contributed by NM, 26-Oct-2007.) |
β’ π· β (CMetβπ) β β’ π· β (Metβπ) | ||
Theorem | cmetcaulem 24812* | Lemma for cmetcau 24813. (Contributed by Mario Carneiro, 14-Oct-2015.) |
β’ π½ = (MetOpenβπ·) & β’ (π β π· β (CMetβπ)) & β’ (π β π β π) & β’ (π β πΉ β (Cauβπ·)) & β’ πΊ = (π₯ β β β¦ if(π₯ β dom πΉ, (πΉβπ₯), π)) β β’ (π β πΉ β dom (βπ‘βπ½)) | ||
Theorem | cmetcau 24813 | The convergence of a Cauchy sequence in a complete metric space. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Oct-2015.) |
β’ π½ = (MetOpenβπ·) β β’ ((π· β (CMetβπ) β§ πΉ β (Cauβπ·)) β πΉ β dom (βπ‘βπ½)) | ||
Theorem | iscmet3lem3 24814* | Lemma for iscmet3 24817. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ π = (β€β₯βπ) β β’ ((π β β€ β§ π β β+) β βπ β π βπ β (β€β₯βπ)((1 / 2)βπ) < π ) | ||
Theorem | iscmet3lem1 24815* | Lemma for iscmet3 24817. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ π = (β€β₯βπ) & β’ π½ = (MetOpenβπ·) & β’ (π β π β β€) & β’ (π β π· β (Metβπ)) & β’ (π β πΉ:πβΆπ) & β’ (π β βπ β β€ βπ’ β (πβπ)βπ£ β (πβπ)(π’π·π£) < ((1 / 2)βπ)) & β’ (π β βπ β π βπ β (π...π)(πΉβπ) β (πβπ)) β β’ (π β πΉ β (Cauβπ·)) | ||
Theorem | iscmet3lem2 24816* | Lemma for iscmet3 24817. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ π = (β€β₯βπ) & β’ π½ = (MetOpenβπ·) & β’ (π β π β β€) & β’ (π β π· β (Metβπ)) & β’ (π β πΉ:πβΆπ) & β’ (π β βπ β β€ βπ’ β (πβπ)βπ£ β (πβπ)(π’π·π£) < ((1 / 2)βπ)) & β’ (π β βπ β π βπ β (π...π)(πΉβπ) β (πβπ)) & β’ (π β πΊ β (Filβπ)) & β’ (π β π:β€βΆπΊ) & β’ (π β πΉ β dom (βπ‘βπ½)) β β’ (π β (π½ fLim πΊ) β β ) | ||
Theorem | iscmet3 24817* | The property "π· is a complete metric" expressed in terms of functions on β (or any other upper integer set). Thus, we only have to look at functions on β, 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.) |
β’ π = (β€β₯βπ) & β’ π½ = (MetOpenβπ·) & β’ (π β π β β€) & β’ (π β π· β (Metβπ)) β β’ (π β (π· β (CMetβπ) β βπ β (Cauβπ·)(π:πβΆπ β π β dom (βπ‘βπ½)))) | ||
Theorem | iscmet2 24818 | A metric π· 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.) |
β’ π½ = (MetOpenβπ·) β β’ (π· β (CMetβπ) β (π· β (Metβπ) β§ (Cauβπ·) β dom (βπ‘βπ½))) | ||
Theorem | cfilresi 24819 | A Cauchy filter on a metric subspace extends to a Cauchy filter in the larger space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ ((π· β (βMetβπ) β§ πΉ β (CauFilβ(π· βΎ (π Γ π)))) β (πfilGenπΉ) β (CauFilβπ·)) | ||
Theorem | cfilres 24820 | Cauchy filter on a metric subspace. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ ((π· β (βMetβπ) β§ πΉ β (Filβπ) β§ π β πΉ) β (πΉ β (CauFilβπ·) β (πΉ βΎt π) β (CauFilβ(π· βΎ (π Γ π))))) | ||
Theorem | caussi 24821 | Cauchy sequence on a metric subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.) |
β’ (π· β (βMetβπ) β (Cauβ(π· βΎ (π Γ π))) β (Cauβπ·)) | ||
Theorem | causs 24822 | Cauchy sequence on a metric subspace. (Contributed by NM, 29-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.) |
β’ ((π· β (βMetβπ) β§ πΉ:ββΆπ) β (πΉ β (Cauβπ·) β πΉ β (Cauβ(π· βΎ (π Γ π))))) | ||
Theorem | equivcfil 24823* | If the metric π· is "strongly finer" than πΆ (meaning that there is a positive real constant π such that πΆ(π₯, π¦) β€ π Β· π·(π₯, π¦)), all the π·-Cauchy filters are also πΆ-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.) |
β’ (π β πΆ β (Metβπ)) & β’ (π β π· β (Metβπ)) & β’ (π β π β β+) & β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯πΆπ¦) β€ (π Β· (π₯π·π¦))) β β’ (π β (CauFilβπ·) β (CauFilβπΆ)) | ||
Theorem | equivcau 24824* | If the metric π· is "strongly finer" than πΆ (meaning that there is a positive real constant π such that πΆ(π₯, π¦) β€ π Β· π·(π₯, π¦)), all the π·-Cauchy sequences are also πΆ-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.) |
β’ (π β πΆ β (Metβπ)) & β’ (π β π· β (Metβπ)) & β’ (π β π β β+) & β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯πΆπ¦) β€ (π Β· (π₯π·π¦))) β β’ (π β (Cauβπ·) β (CauβπΆ)) | ||
Theorem | lmle 24825* | 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.) |
β’ π = (β€β₯βπ) & β’ π½ = (MetOpenβπ·) & β’ (π β π· β (βMetβπ)) & β’ (π β π β β€) & β’ (π β πΉ(βπ‘βπ½)π) & β’ (π β π β π) & β’ (π β π β β*) & β’ ((π β§ π β π) β (ππ·(πΉβπ)) β€ π ) β β’ (π β (ππ·π) β€ π ) | ||
Theorem | nglmle 24826* | If the norm of each member of a converging sequence is less than or equal to a given amount, so is the norm of the convergence value. (Contributed by NM, 25-Dec-2007.) (Revised by AV, 16-Oct-2021.) |
β’ π = (BaseβπΊ) & β’ π· = ((distβπΊ) βΎ (π Γ π)) & β’ π½ = (MetOpenβπ·) & β’ π = (normβπΊ) & β’ (π β πΊ β NrmGrp) & β’ (π β πΉ:ββΆπ) & β’ (π β πΉ(βπ‘βπ½)π) & β’ (π β π β β*) & β’ ((π β§ π β β) β (πβ(πΉβπ)) β€ π ) β β’ (π β (πβπ) β€ π ) | ||
Theorem | lmclim 24827 | 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.) |
β’ π½ = (TopOpenββfld) & β’ π = (β€β₯βπ) β β’ ((π β β€ β§ π β dom πΉ) β (πΉ(βπ‘βπ½)π β (πΉ β (β βpm β) β§ πΉ β π))) | ||
Theorem | lmclimf 24828 | 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.) |
β’ π½ = (TopOpenββfld) & β’ π = (β€β₯βπ) β β’ ((π β β€ β§ πΉ:πβΆβ) β (πΉ(βπ‘βπ½)π β πΉ β π)) | ||
Theorem | metelcls 24829* | 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 10432. 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.) |
β’ π½ = (MetOpenβπ·) & β’ (π β π· β (βMetβπ)) & β’ (π β π β π) β β’ (π β (π β ((clsβπ½)βπ) β βπ(π:ββΆπ β§ π(βπ‘βπ½)π))) | ||
Theorem | metcld 24830* | 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.) |
β’ π½ = (MetOpenβπ·) β β’ ((π· β (βMetβπ) β§ π β π) β (π β (Clsdβπ½) β βπ₯βπ((π:ββΆπ β§ π(βπ‘βπ½)π₯) β π₯ β π))) | ||
Theorem | metcld2 24831 | 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.) |
β’ π½ = (MetOpenβπ·) β β’ ((π· β (βMetβπ) β§ π β π) β (π β (Clsdβπ½) β ((βπ‘βπ½) β (π βm β)) β π)) | ||
Theorem | caubl 24832* | 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.) |
β’ (π β π· β (βMetβπ)) & β’ (π β πΉ:ββΆ(π Γ β+)) & β’ (π β βπ β β ((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ))) & β’ (π β βπ β β+ βπ β β (2nd β(πΉβπ)) < π) β β’ (π β (1st β πΉ) β (Cauβπ·)) | ||
Theorem | caublcls 24833* | 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.) |
β’ (π β π· β (βMetβπ)) & β’ (π β πΉ:ββΆ(π Γ β+)) & β’ (π β βπ β β ((ballβπ·)β(πΉβ(π + 1))) β ((ballβπ·)β(πΉβπ))) & β’ π½ = (MetOpenβπ·) β β’ ((π β§ (1st β πΉ)(βπ‘βπ½)π β§ π΄ β β) β π β ((clsβπ½)β((ballβπ·)β(πΉβπ΄)))) | ||
Theorem | metcnp4 24834* | Two ways to say a mapping from metric πΆ to metric π· is continuous at point π. Theorem 14-4.3 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 4-May-2014.) |
β’ π½ = (MetOpenβπΆ) & β’ πΎ = (MetOpenβπ·) & β’ (π β πΆ β (βMetβπ)) & β’ (π β π· β (βMetβπ)) & β’ (π β π β π) β β’ (π β (πΉ β ((π½ CnP πΎ)βπ) β (πΉ:πβΆπ β§ βπ((π:ββΆπ β§ π(βπ‘βπ½)π) β (πΉ β π)(βπ‘βπΎ)(πΉβπ))))) | ||
Theorem | metcn4 24835* | Two ways to say a mapping from metric πΆ to metric π· is continuous. Theorem 10.3 of [Munkres] p. 128. (Contributed by NM, 13-Jun-2007.) (Revised by Mario Carneiro, 4-May-2014.) |
β’ π½ = (MetOpenβπΆ) & β’ πΎ = (MetOpenβπ·) & β’ (π β πΆ β (βMetβπ)) & β’ (π β π· β (βMetβπ)) & β’ (π β πΉ:πβΆπ) β β’ (π β (πΉ β (π½ Cn πΎ) β βπ(π:ββΆπ β βπ₯(π(βπ‘βπ½)π₯ β (πΉ β π)(βπ‘βπΎ)(πΉβπ₯))))) | ||
Theorem | iscmet3i 24836* | Properties that determine a complete metric space. (Contributed by NM, 15-Apr-2007.) (Revised by Mario Carneiro, 5-May-2014.) |
β’ π½ = (MetOpenβπ·) & β’ π· β (Metβπ) & β’ ((π β (Cauβπ·) β§ π:ββΆπ) β π β dom (βπ‘βπ½)) β β’ π· β (CMetβπ) | ||
Theorem | lmcau 24837 | 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.) |
β’ π½ = (MetOpenβπ·) β β’ (π· β (βMetβπ) β dom (βπ‘βπ½) β (Cauβπ·)) | ||
Theorem | flimcfil 24838 | Every convergent filter in a metric space is a Cauchy filter. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ π½ = (MetOpenβπ·) β β’ ((π· β (βMetβπ) β§ π΄ β (π½ fLim πΉ)) β πΉ β (CauFilβπ·)) | ||
Theorem | metsscmetcld 24839 | A complete subspace of a metric space is closed in the parent space. Formerly part of proof for cmetss 24840. (Contributed by NM, 28-Jan-2008.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 9-Oct-2022.) |
β’ π½ = (MetOpenβπ·) β β’ ((π· β (Metβπ) β§ (π· βΎ (π Γ π)) β (CMetβπ)) β π β (Clsdβπ½)) | ||
Theorem | cmetss 24840 | 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.) (Proof shortened by AV, 9-Oct-2022.) |
β’ π½ = (MetOpenβπ·) β β’ (π· β (CMetβπ) β ((π· βΎ (π Γ π)) β (CMetβπ) β π β (Clsdβπ½))) | ||
Theorem | equivcmet 24841* | If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 24824, metss2 24028, this theorem does not have a one-directional form - it is possible for a metric πΆ that is strongly finer than the complete metric π· to be incomplete and vice versa. Consider π· = the metric on β induced by the usual homeomorphism from (0, 1) against the usual metric πΆ on β and against the discrete metric πΈ on β. Then both πΆ and πΈ are complete but π· is not, and πΆ is strongly finer than π·, which is strongly finer than πΈ. (Contributed by Mario Carneiro, 15-Sep-2015.) |
β’ (π β πΆ β (Metβπ)) & β’ (π β π· β (Metβπ)) & β’ (π β π β β+) & β’ (π β π β β+) & β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯πΆπ¦) β€ (π Β· (π₯π·π¦))) & β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯π·π¦) β€ (π Β· (π₯πΆπ¦))) β β’ (π β (πΆ β (CMetβπ) β π· β (CMetβπ))) | ||
Theorem | relcmpcmet 24842* | If π· is a metric space such that all the balls of some fixed size are relatively compact, then π· is complete. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ π½ = (MetOpenβπ·) & β’ (π β π· β (Metβπ)) & β’ (π β π β β+) & β’ ((π β§ π₯ β π) β (π½ βΎt ((clsβπ½)β(π₯(ballβπ·)π ))) β Comp) β β’ (π β π· β (CMetβπ)) | ||
Theorem | cmpcmet 24843 | A compact metric space is complete. One half of heibor 36781. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ π½ = (MetOpenβπ·) & β’ (π β π· β (Metβπ)) & β’ (π β π½ β Comp) β β’ (π β π· β (CMetβπ)) | ||
Theorem | cfilucfil3 24844 | Given a metric π· 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.) |
β’ ((π β β β§ π· β (βMetβπ)) β ((πΆ β (Filβπ) β§ πΆ β (CauFiluβ(metUnifβπ·))) β πΆ β (CauFilβπ·))) | ||
Theorem | cfilucfil4 24845 | Given a metric π· 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.) |
β’ ((π β β β§ π· β (βMetβπ) β§ πΆ β (Filβπ)) β (πΆ β (CauFiluβ(metUnifβπ·)) β πΆ β (CauFilβπ·))) | ||
Theorem | cncmet 24846 | 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.) |
β’ π· = (abs β β ) β β’ π· β (CMetββ) | ||
Theorem | recmet 24847 | The real numbers are a complete metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
β’ ((abs β β ) βΎ (β Γ β)) β (CMetββ) | ||
Theorem | bcthlem1 24848* | Lemma for bcth 24853. Substitutions for the function πΉ. (Contributed by Mario Carneiro, 9-Jan-2014.) |
β’ π½ = (MetOpenβπ·) & β’ (π β π· β (CMetβπ)) & β’ πΉ = (π β β, π§ β (π Γ β+) β¦ {β¨π₯, πβ© β£ ((π₯ β π β§ π β β+) β§ (π < (1 / π) β§ ((clsβπ½)β(π₯(ballβπ·)π)) β (((ballβπ·)βπ§) β (πβπ))))}) β β’ ((π β§ (π΄ β β β§ π΅ β (π Γ β+))) β (πΆ β (π΄πΉπ΅) β (πΆ β (π Γ β+) β§ (2nd βπΆ) < (1 / π΄) β§ ((clsβπ½)β((ballβπ·)βπΆ)) β (((ballβπ·)βπ΅) β (πβπ΄))))) | ||
Theorem | bcthlem2 24849* | Lemma for bcth 24853. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.) |
β’ π½ = (MetOpenβπ·) & β’ (π β π· β (CMetβπ)) & β’ πΉ = (π β β, π§ β (π Γ β+) β¦ {β¨π₯, πβ© β£ ((π₯ β π β§ π β β+) β§ (π < (1 / π) β§ ((clsβπ½)β(π₯(ballβπ·)π)) β (((ballβπ·)βπ§) β (πβπ))))}) & β’ (π β π:ββΆ(Clsdβπ½)) & β’ (π β π β β+) & β’ (π β πΆ β π) & β’ (π β π:ββΆ(π Γ β+)) & β’ (π β (πβ1) = β¨πΆ, π β©) & β’ (π β βπ β β (πβ(π + 1)) β (ππΉ(πβπ))) β β’ (π β βπ β β ((ballβπ·)β(πβ(π + 1))) β ((ballβπ·)β(πβπ))) | ||
Theorem | bcthlem3 24850* | Lemma for bcth 24853. 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.) |
β’ π½ = (MetOpenβπ·) & β’ (π β π· β (CMetβπ)) & β’ πΉ = (π β β, π§ β (π Γ β+) β¦ {β¨π₯, πβ© β£ ((π₯ β π β§ π β β+) β§ (π < (1 / π) β§ ((clsβπ½)β(π₯(ballβπ·)π)) β (((ballβπ·)βπ§) β (πβπ))))}) & β’ (π β π:ββΆ(Clsdβπ½)) & β’ (π β π β β+) & β’ (π β πΆ β π) & β’ (π β π:ββΆ(π Γ β+)) & β’ (π β (πβ1) = β¨πΆ, π β©) & β’ (π β βπ β β (πβ(π + 1)) β (ππΉ(πβπ))) β β’ ((π β§ (1st β π)(βπ‘βπ½)π₯ β§ π΄ β β) β π₯ β ((ballβπ·)β(πβπ΄))) | ||
Theorem | bcthlem4 24851* | Lemma for bcth 24853. Given any open ball (πΆ(ballβπ·)π ) as starting point (and in particular, a ball in int(βͺ ran π)), the limit point π₯ of the centers of the induced sequence of balls π is outside βͺ ran π. Note that a set π΄ has empty interior iff every nonempty open set π contains points outside π΄, i.e. (π β π΄) β β . (Contributed by Mario Carneiro, 7-Jan-2014.) |
β’ π½ = (MetOpenβπ·) & β’ (π β π· β (CMetβπ)) & β’ πΉ = (π β β, π§ β (π Γ β+) β¦ {β¨π₯, πβ© β£ ((π₯ β π β§ π β β+) β§ (π < (1 / π) β§ ((clsβπ½)β(π₯(ballβπ·)π)) β (((ballβπ·)βπ§) β (πβπ))))}) & β’ (π β π:ββΆ(Clsdβπ½)) & β’ (π β π β β+) & β’ (π β πΆ β π) & β’ (π β π:ββΆ(π Γ β+)) & β’ (π β (πβ1) = β¨πΆ, π β©) & β’ (π β βπ β β (πβ(π + 1)) β (ππΉ(πβπ))) β β’ (π β ((πΆ(ballβπ·)π ) β βͺ ran π) β β ) | ||
Theorem | bcthlem5 24852* |
Lemma for bcth 24853. The proof makes essential use of the Axiom
of
Dependent Choice axdc4uz 13951, which in the form used here accepts a
"selection" function πΉ from each element of πΎ to a
nonempty
subset of πΎ, and the result function π maps
π(π + 1)
to an element of πΉ(π, π(π)). The trick here is thus in
the choice of πΉ and πΎ: we let πΎ 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 πΉ(π, β¨π₯, π§β©) gives the
set of all balls of size less than 1 / π, tagged by their
centers, whose closures fit within the given open set π§ and
miss
π(π).
Since π(π) is closed, π§ β π(π) is open and also nonempty, since π§ is nonempty and π(π) has empty interior. Then there is some ball contained in it, and hence our function πΉ is valid (it never maps to the empty set). Now starting at a point in the interior of βͺ ran π, DC gives us the function π all whose elements are constrained by πΉ acting on the previous value. (This is all proven in this lemma.) Now π is a sequence of tagged open balls, forming an inclusion chain (see bcthlem2 24849) and whose sizes tend to zero, since they are bounded above by 1 / π. Thus, the centers of these balls form a Cauchy sequence, and converge to a point π₯ (see bcthlem4 24851). Since the inclusion chain also ensures the closure of each ball is in the previous ball, the point π₯ must be in all these balls (see bcthlem3 24850) and hence misses each π(π), contradicting the fact that π₯ is in the interior of βͺ ran π (which was the starting point). (Contributed by Mario Carneiro, 6-Jan-2014.) |
β’ π½ = (MetOpenβπ·) & β’ (π β π· β (CMetβπ)) & β’ πΉ = (π β β, π§ β (π Γ β+) β¦ {β¨π₯, πβ© β£ ((π₯ β π β§ π β β+) β§ (π < (1 / π) β§ ((clsβπ½)β(π₯(ballβπ·)π)) β (((ballβπ·)βπ§) β (πβπ))))}) & β’ (π β π:ββΆ(Clsdβπ½)) & β’ (π β βπ β β ((intβπ½)β(πβπ)) = β ) β β’ (π β ((intβπ½)ββͺ ran π) = β ) | ||
Theorem | bcth 24853* | 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 πβπ 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 24852 for an overview of the proof. (Contributed by NM, 28-Oct-2007.) (Proof shortened by Mario Carneiro, 6-Jan-2014.) |
β’ π½ = (MetOpenβπ·) β β’ ((π· β (CMetβπ) β§ π:ββΆ(Clsdβπ½) β§ ((intβπ½)ββͺ ran π) β β ) β βπ β β ((intβπ½)β(πβπ)) β β ) | ||
Theorem | bcth2 24854* | Baire's Category Theorem, version 2: If countably many closed sets cover π, then one of them has an interior. (Contributed by Mario Carneiro, 10-Jan-2014.) |
β’ π½ = (MetOpenβπ·) β β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β βπ β β ((intβπ½)β(πβπ)) β β ) | ||
Theorem | bcth3 24855* | Baire's Category Theorem, version 3: The intersection of countably many dense open sets is dense. (Contributed by Mario Carneiro, 10-Jan-2014.) |
β’ π½ = (MetOpenβπ·) β β’ ((π· β (CMetβπ) β§ π:ββΆπ½ β§ βπ β β ((clsβπ½)β(πβπ)) = π) β ((clsβπ½)ββ© ran π) = π) | ||
Syntax | ccms 24856 | Extend class notation with the class of complete metric spaces. |
class CMetSp | ||
Syntax | cbn 24857 | Extend class notation with the class of Banach spaces. |
class Ban | ||
Syntax | chl 24858 | Extend class notation with the class of subcomplex Hilbert spaces. |
class βHil | ||
Definition | df-cms 24859* | Define the class of complete metric spaces. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ CMetSp = {π€ β MetSp β£ [(Baseβπ€) / π]((distβπ€) βΎ (π Γ π)) β (CMetβπ)} | ||
Definition | df-bn 24860 | 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.) |
β’ Ban = {π€ β (NrmVec β© CMetSp) β£ (Scalarβπ€) β CMetSp} | ||
Definition | df-hl 24861 | 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 subfield of the field of complex numbers closed under square roots of nonnegative reals. (Contributed by Steve Rodriguez, 28-Apr-2007.) |
β’ βHil = (Ban β© βPreHil) | ||
Theorem | isbn 24862 | 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.) |
β’ πΉ = (Scalarβπ) β β’ (π β Ban β (π β NrmVec β§ π β CMetSp β§ πΉ β CMetSp)) | ||
Theorem | bnsca 24863 | The scalar field of a Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.) |
β’ πΉ = (Scalarβπ) β β’ (π β Ban β πΉ β CMetSp) | ||
Theorem | bnnvc 24864 | A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ (π β Ban β π β NrmVec) | ||
Theorem | bnnlm 24865 | A Banach space is a normed module. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ (π β Ban β π β NrmMod) | ||
Theorem | bnngp 24866 | A Banach space is a normed group. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ (π β Ban β π β NrmGrp) | ||
Theorem | bnlmod 24867 | A Banach space is a left module. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ (π β Ban β π β LMod) | ||
Theorem | bncms 24868 | A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ (π β Ban β π β CMetSp) | ||
Theorem | iscms 24869 | A complete metric space is a metric space with a complete metric. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ π = (Baseβπ) & β’ π· = ((distβπ) βΎ (π Γ π)) β β’ (π β CMetSp β (π β MetSp β§ π· β (CMetβπ))) | ||
Theorem | cmscmet 24870 | The induced metric on a complete normed group is complete. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ π = (Baseβπ) & β’ π· = ((distβπ) βΎ (π Γ π)) β β’ (π β CMetSp β π· β (CMetβπ)) | ||
Theorem | bncmet 24871 | The induced metric on Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.) |
β’ π = (Baseβπ) & β’ π· = ((distβπ) βΎ (π Γ π)) β β’ (π β Ban β π· β (CMetβπ)) | ||
Theorem | cmsms 24872 | A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ (πΊ β CMetSp β πΊ β MetSp) | ||
Theorem | cmspropd 24873 | Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ (π β π΅ = (BaseβπΎ)) & β’ (π β π΅ = (BaseβπΏ)) & β’ (π β ((distβπΎ) βΎ (π΅ Γ π΅)) = ((distβπΏ) βΎ (π΅ Γ π΅))) & β’ (π β (TopOpenβπΎ) = (TopOpenβπΏ)) β β’ (π β (πΎ β CMetSp β πΏ β CMetSp)) | ||
Theorem | cmssmscld 24874 | The restriction of a metric space is closed if it is complete. (Contributed by AV, 9-Oct-2022.) |
β’ πΎ = (π βΎs π΄) & β’ π = (Baseβπ) & β’ π½ = (TopOpenβπ) β β’ ((π β MetSp β§ π΄ β π β§ πΎ β CMetSp) β π΄ β (Clsdβπ½)) | ||
Theorem | cmsss 24875 | The restriction of a complete metric space is complete iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ πΎ = (π βΎs π΄) & β’ π = (Baseβπ) & β’ π½ = (TopOpenβπ) β β’ ((π β CMetSp β§ π΄ β π) β (πΎ β CMetSp β π΄ β (Clsdβπ½))) | ||
Theorem | lssbn 24876 | A subspace of a Banach space is a Banach space iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ π = (π βΎs π) & β’ π = (LSubSpβπ) & β’ π½ = (TopOpenβπ) β β’ ((π β Ban β§ π β π) β (π β Ban β π β (Clsdβπ½))) | ||
Theorem | cmetcusp1 24877 | 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.) |
β’ π = (BaseβπΉ) & β’ π· = ((distβπΉ) βΎ (π Γ π)) & β’ π = (UnifStβπΉ) β β’ ((π β β β§ πΉ β CMetSp β§ π = (metUnifβπ·)) β πΉ β CUnifSp) | ||
Theorem | cmetcusp 24878 | The uniform space generated by a complete metric is a complete uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.) |
β’ ((π β β β§ π· β (CMetβπ)) β (toUnifSpβ(metUnifβπ·)) β CUnifSp) | ||
Theorem | cncms 24879 | The field of complex numbers is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ βfld β CMetSp | ||
Theorem | cnflduss 24880 | The uniform structure of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
β’ π = (UnifStββfld) β β’ π = (metUnifβ(abs β β )) | ||
Theorem | cnfldcusp 24881 | The field of complex numbers is a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.) |
β’ βfld β CUnifSp | ||
Theorem | resscdrg 24882 | The real numbers are a subset of any complete subfield in the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ πΉ = (βfld βΎs πΎ) β β’ ((πΎ β (SubRingββfld) β§ πΉ β DivRing β§ πΉ β CMetSp) β β β πΎ) | ||
Theorem | cncdrg 24883 | The only complete subfields of the complex numbers are β and β. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ πΉ = (βfld βΎs πΎ) β β’ ((πΎ β (SubRingββfld) β§ πΉ β DivRing β§ πΉ β CMetSp) β πΎ β {β, β}) | ||
Theorem | srabn 24884 | 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.) |
β’ π΄ = ((subringAlg βπ)βπ) & β’ π½ = (TopOpenβπ) β β’ ((π β NrmRing β§ π β CMetSp β§ π β (SubRingβπ)) β (π΄ β Ban β (π β (Clsdβπ½) β§ (π βΎs π) β DivRing))) | ||
Theorem | rlmbn 24885 | The ring module over a complete normed division ring is a Banach space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ ((π β NrmRing β§ π β DivRing β§ π β CMetSp) β (ringLModβπ ) β Ban) | ||
Theorem | ishl 24886 | 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.) |
β’ (π β βHil β (π β Ban β§ π β βPreHil)) | ||
Theorem | hlbn 24887 | Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) |
β’ (π β βHil β π β Ban) | ||
Theorem | hlcph 24888 | Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ (π β βHil β π β βPreHil) | ||
Theorem | hlphl 24889 | 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.) |
β’ (π β βHil β π β PreHil) | ||
Theorem | hlcms 24890 | Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.) |
β’ (π β βHil β π β CMetSp) | ||
Theorem | hlprlem 24891 | Lemma for hlpr 24893. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) β β’ (π β βHil β (πΎ β (SubRingββfld) β§ (βfld βΎs πΎ) β DivRing β§ (βfld βΎs πΎ) β CMetSp)) | ||
Theorem | hlress 24892 | The scalar field of a subcomplex Hilbert space contains β. (Contributed by Mario Carneiro, 8-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) β β’ (π β βHil β β β πΎ) | ||
Theorem | hlpr 24893 | The scalar field of a subcomplex Hilbert space is either β or β. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) β β’ (π β βHil β πΎ β {β, β}) | ||
Theorem | ishl2 24894 | A Hilbert space is a complete subcomplex pre-Hilbert space over β or β. (Contributed by Mario Carneiro, 15-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ πΎ = (BaseβπΉ) β β’ (π β βHil β (π β CMetSp β§ π β βPreHil β§ πΎ β {β, β})) | ||
Theorem | cphssphl 24895 | A Banach subspace of a subcomplex pre-Hilbert space is a subcomplex Hilbert space. (Contributed by NM, 11-Apr-2008.) (Revised by AV, 25-Sep-2022.) |
β’ π = (π βΎs π) & β’ π = (LSubSpβπ) β β’ ((π β βPreHil β§ π β π β§ π β Ban) β π β βHil) | ||
Theorem | cmslssbn 24896 | A complete linear subspace of a normed vector space is a Banach space. We furthermore have to assume that the field of scalars is complete since this is a requirement in the current definition of Banach spaces df-bn 24860. (Contributed by AV, 8-Oct-2022.) |
β’ π = (π βΎs π) & β’ π = (LSubSpβπ) β β’ (((π β NrmVec β§ (Scalarβπ) β CMetSp) β§ (π β CMetSp β§ π β π)) β π β Ban) | ||
Theorem | cmscsscms 24897 | A closed subspace of a complete metric space which is also a subcomplex pre-Hilbert space is a complete metric space. Remark: the assumption that the Banach space must be a (subcomplex) pre-Hilbert space is required because the definition of ClSubSp is based on an inner product. If ClSubSp was generalized to arbitrary topological spaces (or at least topological modules), this assumption could be omitted. (Contributed by AV, 8-Oct-2022.) |
β’ π = (π βΎs π) & β’ π = (ClSubSpβπ) β β’ (((π β CMetSp β§ π β βPreHil) β§ π β π) β π β CMetSp) | ||
Theorem | bncssbn 24898 | A closed subspace of a Banach space which is also a subcomplex pre-Hilbert space is a Banach space. Remark: the assumption that the Banach space must be a (subcomplex) pre-Hilbert space is required because the definition of ClSubSp is based on an inner product. If ClSubSp was generalized for arbitrary topological spaces, this assuption could be omitted. (Contributed by AV, 8-Oct-2022.) |
β’ π = (π βΎs π) & β’ π = (ClSubSpβπ) β β’ (((π β Ban β§ π β βPreHil) β§ π β π) β π β Ban) | ||
Theorem | cssbn 24899 | A complete subspace of a normed vector space with a complete scalar field is a Banach space. Remark: In contrast to ClSubSp, a complete subspace is defined by "a linear subspace in which all Cauchy sequences converge to a point in the subspace". This is closer to the original, but deprecated definition Cβ (df-ch 30512) of closed subspaces of a Hilbert space. It may be superseded by cmslssbn 24896. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 6-Oct-2022.) |
β’ π = (π βΎs π) & β’ π = (LSubSpβπ) & β’ π· = ((distβπ) βΎ (π Γ π)) β β’ (((π β NrmVec β§ (Scalarβπ) β CMetSp β§ π β π) β§ (Cauβπ·) β dom (βπ‘β(MetOpenβπ·))) β π β Ban) | ||
Theorem | csschl 24900 | A complete subspace of a complex pre-Hilbert space is a complex Hilbert space. Remarks: (a) In contrast to ClSubSp, a complete subspace is defined by "a linear subspace in which all Cauchy sequences converge to a point in the subspace". This is closer to the original, but deprecated definition Cβ (df-ch 30512) of closed subspaces of a Hilbert space. (b) This theorem does not hold for arbitrary subcomplex (pre-)Hilbert spaces, because the scalar field as restriction of the field of the complex numbers need not be closed. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 6-Oct-2022.) |
β’ π = (π βΎs π) & β’ π = (LSubSpβπ) & β’ π· = ((distβπ) βΎ (π Γ π)) & β’ (Scalarβπ) = βfld β β’ ((π β βPreHil β§ π β π β§ (Cauβπ·) β dom (βπ‘β(MetOpenβπ·))) β (π β βHil β§ (Scalarβπ) = βfld)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |