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Theorem List for Metamath Proof Explorer - 23601-23700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcaucfil 23601 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.)
𝑍 = (ℤ𝑀)    &   𝐿 = ((𝑋 FilMap 𝐹)‘(ℤ𝑍))       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐿 ∈ (CauFil‘𝐷)))

Theoremiscmet 23602* The property "𝐷 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.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))

Theoremcmetcvg 23603 The convergence of a Cauchy filter in a complete metric space. (Contributed by Mario Carneiro, 14-Oct-2015.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝐹) ≠ ∅)

Theoremcmetmet 23604 A complete metric space is a metric space. (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 29-Jan-2014.)
(𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))

Theoremcmetmeti 23605 A complete metric space is a metric space. (Contributed by NM, 26-Oct-2007.)
𝐷 ∈ (CMet‘𝑋)       𝐷 ∈ (Met‘𝑋)

Theoremcmetcaulem 23606* Lemma for cmetcau 23607. (Contributed by Mario Carneiro, 14-Oct-2015.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   (𝜑𝑃𝑋)    &   (𝜑𝐹 ∈ (Cau‘𝐷))    &   𝐺 = (𝑥 ∈ ℕ ↦ if(𝑥 ∈ dom 𝐹, (𝐹𝑥), 𝑃))       (𝜑𝐹 ∈ dom (⇝𝑡𝐽))

Theoremcmetcau 23607 The convergence of a Cauchy sequence in a complete metric space. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Oct-2015.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ dom (⇝𝑡𝐽))

Theoremiscmet3lem3 23608* Lemma for iscmet3 23611. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝑅 ∈ ℝ+) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((1 / 2)↑𝑘) < 𝑅)

Theoremiscmet3lem1 23609* Lemma for iscmet3 23611. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐹:𝑍𝑋)    &   (𝜑 → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑆𝑘)∀𝑣 ∈ (𝑆𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘))    &   (𝜑 → ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝐹𝑘) ∈ (𝑆𝑛))       (𝜑𝐹 ∈ (Cau‘𝐷))

Theoremiscmet3lem2 23610* Lemma for iscmet3 23611. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐹:𝑍𝑋)    &   (𝜑 → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑆𝑘)∀𝑣 ∈ (𝑆𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘))    &   (𝜑 → ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝐹𝑘) ∈ (𝑆𝑛))    &   (𝜑𝐺 ∈ (Fil‘𝑋))    &   (𝜑𝑆:ℤ⟶𝐺)    &   (𝜑𝐹 ∈ dom (⇝𝑡𝐽))       (𝜑 → (𝐽 fLim 𝐺) ≠ ∅)

Theoremiscmet3 23611* The property "𝐷 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.)
𝑍 = (ℤ𝑀)    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ (Met‘𝑋))       (𝜑 → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))))

Theoremiscmet2 23612 A metric 𝐷 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.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡𝐽)))

Theoremcfilresi 23613 A Cauchy filter on a metric subspace extends to a Cauchy filter in the larger space. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝑋filGen𝐹) ∈ (CauFil‘𝐷))

Theoremcfilres 23614 Cauchy filter on a metric subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))))

Theoremcaussi 23615 Cauchy sequence on a metric subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
(𝐷 ∈ (∞Met‘𝑋) → (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) ⊆ (Cau‘𝐷))

Theoremcauss 23616 Cauchy sequence on a metric subspace. (Contributed by NM, 29-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))))

Theoremequivcfil 23617* If the metric 𝐷 is "strongly finer" than 𝐶 (meaning that there is a positive real constant 𝑅 such that 𝐶(𝑥, 𝑦) ≤ 𝑅 · 𝐷(𝑥, 𝑦)), all the 𝐷-Cauchy filters are also 𝐶-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.)
(𝜑𝐶 ∈ (Met‘𝑋))    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))       (𝜑 → (CauFil‘𝐷) ⊆ (CauFil‘𝐶))

Theoremequivcau 23618* If the metric 𝐷 is "strongly finer" than 𝐶 (meaning that there is a positive real constant 𝑅 such that 𝐶(𝑥, 𝑦) ≤ 𝑅 · 𝐷(𝑥, 𝑦)), all the 𝐷-Cauchy sequences are also 𝐶-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.)
(𝜑𝐶 ∈ (Met‘𝑋))    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))       (𝜑 → (Cau‘𝐷) ⊆ (Cau‘𝐶))

Theoremlmle 23619* 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.)
𝑍 = (ℤ𝑀)    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹(⇝𝑡𝐽)𝑃)    &   (𝜑𝑄𝑋)    &   (𝜑𝑅 ∈ ℝ*)    &   ((𝜑𝑘𝑍) → (𝑄𝐷(𝐹𝑘)) ≤ 𝑅)       (𝜑 → (𝑄𝐷𝑃) ≤ 𝑅)

Theoremnglmle 23620* 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.)
𝑋 = (Base‘𝐺)    &   𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋))    &   𝐽 = (MetOpen‘𝐷)    &   𝑁 = (norm‘𝐺)    &   (𝜑𝐺 ∈ NrmGrp)    &   (𝜑𝐹:ℕ⟶𝑋)    &   (𝜑𝐹(⇝𝑡𝐽)𝑃)    &   (𝜑𝑅 ∈ ℝ*)    &   ((𝜑𝑘 ∈ ℕ) → (𝑁‘(𝐹𝑘)) ≤ 𝑅)       (𝜑 → (𝑁𝑃) ≤ 𝑅)

Theoremlmclim 23621 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.)
𝐽 = (TopOpen‘ℂfld)    &   𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝑍 ⊆ dom 𝐹) → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐹𝑃)))

Theoremlmclimf 23622 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.)
𝐽 = (TopOpen‘ℂfld)    &   𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹(⇝𝑡𝐽)𝑃𝐹𝑃))

Theoremmetelcls 23623* 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 9653. 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.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆𝑋)       (𝜑 → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)))

Theoremmetcld 23624* 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.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ∀𝑥𝑓((𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑥) → 𝑥𝑆)))

Theoremmetcld2 23625 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.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((⇝𝑡𝐽) “ (𝑆𝑚 ℕ)) ⊆ 𝑆))

Theoremcaubl 23626* 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.)
(𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))    &   (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))    &   (𝜑 → ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ (2nd ‘(𝐹𝑛)) < 𝑟)       (𝜑 → (1st𝐹) ∈ (Cau‘𝐷))

Theoremcaublcls 23627* 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.)
(𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))    &   (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))    &   𝐽 = (MetOpen‘𝐷)       ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹𝐴))))

Theoremmetcnp4 23628* Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. Theorem 14-4.3 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 4-May-2014.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   (𝜑𝐶 ∈ (∞Met‘𝑋))    &   (𝜑𝐷 ∈ (∞Met‘𝑌))    &   (𝜑𝑃𝑋)       (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋𝑓(⇝𝑡𝐽)𝑃) → (𝐹𝑓)(⇝𝑡𝐾)(𝐹𝑃)))))

Theoremmetcn4 23629* Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous. Theorem 10.3 of [Munkres] p. 128. (Contributed by NM, 13-Jun-2007.) (Revised by Mario Carneiro, 4-May-2014.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   (𝜑𝐶 ∈ (∞Met‘𝑋))    &   (𝜑𝐷 ∈ (∞Met‘𝑌))    &   (𝜑𝐹:𝑋𝑌)       (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑓(𝑓:ℕ⟶𝑋 → ∀𝑥(𝑓(⇝𝑡𝐽)𝑥 → (𝐹𝑓)(⇝𝑡𝐾)(𝐹𝑥)))))

Theoremiscmet3i 23630* Properties that determine a complete metric space. (Contributed by NM, 15-Apr-2007.) (Revised by Mario Carneiro, 5-May-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝐷 ∈ (Met‘𝑋)    &   ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝑋) → 𝑓 ∈ dom (⇝𝑡𝐽))       𝐷 ∈ (CMet‘𝑋)

Theoremlmcau 23631 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.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → dom (⇝𝑡𝐽) ⊆ (Cau‘𝐷))

Theoremflimcfil 23632 Every convergent filter in a metric space is a Cauchy filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ (𝐽 fLim 𝐹)) → 𝐹 ∈ (CauFil‘𝐷))

Theoremmetsscmetcld 23633 A complete subspace of a metric space is closed in the parent space. Formerly part of proof for cmetss 23634. (Contributed by NM, 28-Jan-2008.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised by AV, 9-Oct-2022.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (Met‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝑌 ∈ (Clsd‘𝐽))

Theoremcmetss 23634 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.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (CMet‘𝑋) → ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ↔ 𝑌 ∈ (Clsd‘𝐽)))

Theoremequivcmet 23635* If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 23618, metss2 22837, this theorem does not have a one-directional form - it is possible for a metric 𝐶 that is strongly finer than the complete metric 𝐷 to be incomplete and vice versa. Consider 𝐷 = the metric on induced by the usual homeomorphism from (0, 1) against the usual metric 𝐶 on and against the discrete metric 𝐸 on . Then both 𝐶 and 𝐸 are complete but 𝐷 is not, and 𝐶 is strongly finer than 𝐷, which is strongly finer than 𝐸. (Contributed by Mario Carneiro, 15-Sep-2015.)
(𝜑𝐶 ∈ (Met‘𝑋))    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝑆 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐷𝑦) ≤ (𝑆 · (𝑥𝐶𝑦)))       (𝜑 → (𝐶 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ (CMet‘𝑋)))

Theoremrelcmpcmet 23636* If 𝐷 is a metric space such that all the balls of some fixed size are relatively compact, then 𝐷 is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑𝑥𝑋) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)       (𝜑𝐷 ∈ (CMet‘𝑋))

Theoremcmpcmet 23637 A compact metric space is complete. One half of heibor 34570. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)       (𝜑𝐷 ∈ (CMet‘𝑋))

Theoremcfilucfil3 23638 Given a metric 𝐷 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.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFilu‘(metUnif‘𝐷))) ↔ 𝐶 ∈ (CauFil‘𝐷)))

Theoremcfilucfil4 23639 Given a metric 𝐷 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.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋) ∧ 𝐶 ∈ (Fil‘𝑋)) → (𝐶 ∈ (CauFilu‘(metUnif‘𝐷)) ↔ 𝐶 ∈ (CauFil‘𝐷)))

Theoremcncmet 23640 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.)
𝐷 = (abs ∘ − )       𝐷 ∈ (CMet‘ℂ)

Theoremrecmet 23641 The real numbers are a complete metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (CMet‘ℝ)

12.5.6  Baire's Category Theorem

Theorembcthlem1 23642* Lemma for bcth 23647. Substitutions for the function 𝐹. (Contributed by Mario Carneiro, 9-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})       ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ+))) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ (𝐶 ∈ (𝑋 × ℝ+) ∧ (2nd𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀𝐴)))))

Theorembcthlem2 23643* Lemma for bcth 23647. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})    &   (𝜑𝑀:ℕ⟶(Clsd‘𝐽))    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐶𝑋)    &   (𝜑𝑔:ℕ⟶(𝑋 × ℝ+))    &   (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)    &   (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)))       (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔𝑛)))

Theorembcthlem3 23644* Lemma for bcth 23647. 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.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})    &   (𝜑𝑀:ℕ⟶(Clsd‘𝐽))    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐶𝑋)    &   (𝜑𝑔:ℕ⟶(𝑋 × ℝ+))    &   (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)    &   (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)))       ((𝜑 ∧ (1st𝑔)(⇝𝑡𝐽)𝑥𝐴 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔𝐴)))

Theorembcthlem4 23645* Lemma for bcth 23647. Given any open ball (𝐶(ball‘𝐷)𝑅) as starting point (and in particular, a ball in int( ran 𝑀)), the limit point 𝑥 of the centers of the induced sequence of balls 𝑔 is outside ran 𝑀. Note that a set 𝐴 has empty interior iff every nonempty open set 𝑈 contains points outside 𝐴, i.e. (𝑈𝐴) ≠ ∅. (Contributed by Mario Carneiro, 7-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})    &   (𝜑𝑀:ℕ⟶(Clsd‘𝐽))    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐶𝑋)    &   (𝜑𝑔:ℕ⟶(𝑋 × ℝ+))    &   (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)    &   (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘)))       (𝜑 → ((𝐶(ball‘𝐷)𝑅) ∖ ran 𝑀) ≠ ∅)

Theorembcthlem5 23646* Lemma for bcth 23647. The proof makes essential use of the Axiom of Dependent Choice axdc4uz 13165, which in the form used here accepts a "selection" function 𝐹 from each element of 𝐾 to a nonempty subset of 𝐾, and the result function 𝑔 maps 𝑔(𝑛 + 1) to an element of 𝐹(𝑛, 𝑔(𝑛)). The trick here is thus in the choice of 𝐹 and 𝐾: we let 𝐾 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 𝐹(𝑘, ⟨𝑥, 𝑧⟩) gives the set of all balls of size less than 1 / 𝑘, tagged by their centers, whose closures fit within the given open set 𝑧 and miss 𝑀(𝑘).

Since 𝑀(𝑘) is closed, 𝑧𝑀(𝑘) is open and also nonempty, since 𝑧 is nonempty and 𝑀(𝑘) has empty interior. Then there is some ball contained in it, and hence our function 𝐹 is valid (it never maps to the empty set). Now starting at a point in the interior of ran 𝑀, DC gives us the function 𝑔 all whose elements are constrained by 𝐹 acting on the previous value. (This is all proven in this lemma.) Now 𝑔 is a sequence of tagged open balls, forming an inclusion chain (see bcthlem2 23643) and whose sizes tend to zero, since they are bounded above by 1 / 𝑘. Thus, the centers of these balls form a Cauchy sequence, and converge to a point 𝑥 (see bcthlem4 23645). Since the inclusion chain also ensures the closure of each ball is in the previous ball, the point 𝑥 must be in all these balls (see bcthlem3 23644) and hence misses each 𝑀(𝑘), contradicting the fact that 𝑥 is in the interior of ran 𝑀 (which was the starting point). (Contributed by Mario Carneiro, 6-Jan-2014.)

𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝑋𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀𝑘))))})    &   (𝜑𝑀:ℕ⟶(Clsd‘𝐽))    &   (𝜑 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) = ∅)       (𝜑 → ((int‘𝐽)‘ ran 𝑀) = ∅)

Theorembcth 23647* 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 𝑀𝑘 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 23646 for an overview of the proof. (Contributed by NM, 28-Oct-2007.) (Proof shortened by Mario Carneiro, 6-Jan-2014.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶(Clsd‘𝐽) ∧ ((int‘𝐽)‘ ran 𝑀) ≠ ∅) → ∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅)

Theorembcth2 23648* Baire's Category Theorem, version 2: If countably many closed sets cover 𝑋, then one of them has an interior. (Contributed by Mario Carneiro, 10-Jan-2014.)
𝐽 = (MetOpen‘𝐷)       (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑋 ≠ ∅) ∧ (𝑀:ℕ⟶(Clsd‘𝐽) ∧ ran 𝑀 = 𝑋)) → ∃𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀𝑘)) ≠ ∅)

Theorembcth3 23649* Baire's Category Theorem, version 3: The intersection of countably many dense open sets is dense. (Contributed by Mario Carneiro, 10-Jan-2014.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑀:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ ((cls‘𝐽)‘(𝑀𝑘)) = 𝑋) → ((cls‘𝐽)‘ ran 𝑀) = 𝑋)

12.5.7  Banach spaces and subcomplex Hilbert spaces

Syntaxccms 23650 Extend class notation with the class of complete metric spaces.
class CMetSp

Syntaxcbn 23651 Extend class notation with the class of Banach spaces.
class Ban

Syntaxchl 23652 Extend class notation with the class of subcomplex Hilbert spaces.
class ℂHil

Definitiondf-cms 23653* Define the class of complete metric spaces. (Contributed by Mario Carneiro, 15-Oct-2015.)
CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}

Definitiondf-bn 23654 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 = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp}

Definitiondf-hl 23655 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.)
ℂHil = (Ban ∩ ℂPreHil)

Theoremisbn 23656 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.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))

Theorembnsca 23657 The scalar field of a Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ Ban → 𝐹 ∈ CMetSp)

Theorembnnvc 23658 A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑊 ∈ Ban → 𝑊 ∈ NrmVec)

Theorembnnlm 23659 A Banach space is a normed module. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑊 ∈ Ban → 𝑊 ∈ NrmMod)

Theorembnngp 23660 A Banach space is a normed group. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑊 ∈ Ban → 𝑊 ∈ NrmGrp)

Theorembnlmod 23661 A Banach space is a left module. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑊 ∈ Ban → 𝑊 ∈ LMod)

Theorembncms 23662 A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑊 ∈ Ban → 𝑊 ∈ CMetSp)

Theoremiscms 23663 A complete metric space is a metric space with a complete metric. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑀)    &   𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))       (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))

Theoremcmscmet 23664 The induced metric on a complete normed group is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑀)    &   𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))       (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))

Theorembncmet 23665 The induced metric on Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝑋 = (Base‘𝑀)    &   𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))       (𝑀 ∈ Ban → 𝐷 ∈ (CMet‘𝑋))

Theoremcmsms 23666 A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝐺 ∈ CMetSp → 𝐺 ∈ MetSp)

Theoremcmspropd 23667 Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    &   (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))       (𝜑 → (𝐾 ∈ CMetSp ↔ 𝐿 ∈ CMetSp))

Theoremcmssmscld 23668 The restriction of a metric space is closed if it is complete. (Contributed by AV, 9-Oct-2022.)
𝐾 = (𝑀s 𝐴)    &   𝑋 = (Base‘𝑀)    &   𝐽 = (TopOpen‘𝑀)       ((𝑀 ∈ MetSp ∧ 𝐴𝑋𝐾 ∈ CMetSp) → 𝐴 ∈ (Clsd‘𝐽))

Theoremcmsss 23669 The restriction of a complete metric space is complete iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐾 = (𝑀s 𝐴)    &   𝑋 = (Base‘𝑀)    &   𝐽 = (TopOpen‘𝑀)       ((𝑀 ∈ CMetSp ∧ 𝐴𝑋) → (𝐾 ∈ CMetSp ↔ 𝐴 ∈ (Clsd‘𝐽)))

Theoremlssbn 23670 A subspace of a Banach space is a Banach space iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)    &   𝐽 = (TopOpen‘𝑊)       ((𝑊 ∈ Ban ∧ 𝑈𝑆) → (𝑋 ∈ Ban ↔ 𝑈 ∈ (Clsd‘𝐽)))

Theoremcmetcusp1 23671 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.)
𝑋 = (Base‘𝐹)    &   𝐷 = ((dist‘𝐹) ↾ (𝑋 × 𝑋))    &   𝑈 = (UnifSt‘𝐹)       ((𝑋 ≠ ∅ ∧ 𝐹 ∈ CMetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐹 ∈ CUnifSp)

Theoremcmetcusp 23672 The uniform space generated by a complete metric is a complete uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (CMet‘𝑋)) → (toUnifSp‘(metUnif‘𝐷)) ∈ CUnifSp)

Theoremcncms 23673 The field of complex numbers is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
fld ∈ CMetSp

Theoremcnflduss 23674 The uniform structure of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
𝑈 = (UnifSt‘ℂfld)       𝑈 = (metUnif‘(abs ∘ − ))

Theoremcnfldcusp 23675 The field of complex numbers is a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
fld ∈ CUnifSp

Theoremresscdrg 23676 The real numbers are a subset of any complete subfield in the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (ℂflds 𝐾)       ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → ℝ ⊆ 𝐾)

Theoremcncdrg 23677 The only complete subfields of the complex numbers are and . (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (ℂflds 𝐾)       ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → 𝐾 ∈ {ℝ, ℂ})

Theoremsrabn 23678 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.)
𝐴 = ((subringAlg ‘𝑊)‘𝑆)    &   𝐽 = (TopOpen‘𝑊)       ((𝑊 ∈ NrmRing ∧ 𝑊 ∈ CMetSp ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ Ban ↔ (𝑆 ∈ (Clsd‘𝐽) ∧ (𝑊s 𝑆) ∈ DivRing)))

Theoremrlmbn 23679 The ring module over a complete normed division ring is a Banach space. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing ∧ 𝑅 ∈ CMetSp) → (ringLMod‘𝑅) ∈ Ban)

Theoremishl 23680 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.)
(𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))

Theoremhlbn 23681 Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
(𝑊 ∈ ℂHil → 𝑊 ∈ Ban)

Theoremhlcph 23682 Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)

Theoremhlphl 23683 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.)
(𝑊 ∈ ℂHil → 𝑊 ∈ PreHil)

Theoremhlcms 23684 Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.)
(𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)

Theoremhlprlem 23685 Lemma for hlpr 23687. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂHil → (𝐾 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝐾) ∈ DivRing ∧ (ℂflds 𝐾) ∈ CMetSp))

Theoremhlress 23686 The scalar field of a subcomplex Hilbert space contains . (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂHil → ℝ ⊆ 𝐾)

Theoremhlpr 23687 The scalar field of a subcomplex Hilbert space is either or . (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂHil → 𝐾 ∈ {ℝ, ℂ})

Theoremishl2 23688 A Hilbert space is a complete subcomplex pre-Hilbert space over or . (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂHil ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ}))

Theoremcphssphl 23689 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.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝑈𝑆𝑋 ∈ Ban) → 𝑋 ∈ ℂHil)

Theoremcmslssbn 23690 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 23654. (Contributed by AV, 8-Oct-2022.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)       (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp) ∧ (𝑋 ∈ CMetSp ∧ 𝑈𝑆)) → 𝑋 ∈ Ban)

Theoremcmscsscms 23691 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.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (ClSubSp‘𝑊)       (((𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil) ∧ 𝑈𝑆) → 𝑋 ∈ CMetSp)

Theorembncssbn 23692 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.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (ClSubSp‘𝑊)       (((𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil) ∧ 𝑈𝑆) → 𝑋 ∈ Ban)

Theoremcssbn 23693 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 C (df-ch 28789) of closed subspaces of a Hilbert space. It may be superseded by cmslssbn 23690. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 6-Oct-2022.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)    &   𝐷 = ((dist‘𝑊) ↾ (𝑈 × 𝑈))       (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ Ban)

Theoremcsschl 23694 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 C (df-ch 28789) 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.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)    &   𝐷 = ((dist‘𝑊) ↾ (𝑈 × 𝑈))    &   (Scalar‘𝑊) = ℂfld       ((𝑊 ∈ ℂPreHil ∧ 𝑈𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (𝑋 ∈ ℂHil ∧ (Scalar‘𝑋) = ℂfld))

Theoremcmslsschl 23695 A complete linear subspace of a subcomplex Hilbert space is a subcomplex Hilbert space. (Contributed by AV, 8-Oct-2022.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ ℂHil ∧ 𝑋 ∈ CMetSp ∧ 𝑈𝑆) → 𝑋 ∈ ℂHil)

Theoremchlcsschl 23696 A closed subspace of a subcomplex Hilbert space is a subcomplex Hilbert space. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 8-Oct-2022.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (ClSubSp‘𝑊)       ((𝑊 ∈ ℂHil ∧ 𝑈𝑆) → 𝑋 ∈ ℂHil)

12.5.7.1  The complete ordered field of the real numbers

Theoremretopn 23697 The topology of the real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
(topGen‘ran (,)) = (TopOpen‘ℝfld)

Theoremrecms 23698 The real numbers form a complete metric space. (Contributed by Thierry Arnoux, 1-Nov-2017.)
fld ∈ CMetSp

Theoremreust 23699 The Uniform structure of the real numbers. (Contributed by Thierry Arnoux, 14-Feb-2018.)
(UnifSt‘ℝfld) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ)))

Theoremrecusp 23700 The real numbers form a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.)
fld ∈ CUnifSp

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