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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cfilresi 23901 | 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)) | ||
Theorem | cfilres 23902 | Cauchy filter on a metric subspace. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ↾t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))) | ||
Theorem | caussi 23903 | 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)) | ||
Theorem | causs 23904 | 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))))) | ||
Theorem | equivcfil 23905* | 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.) |
⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦))) ⇒ ⊢ (𝜑 → (CauFil‘𝐷) ⊆ (CauFil‘𝐶)) | ||
Theorem | equivcau 23906* | 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.) |
⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦))) ⇒ ⊢ (𝜑 → (Cau‘𝐷) ⊆ (Cau‘𝐶)) | ||
Theorem | lmle 23907* | 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 = (ℤ≥‘𝑀) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) & ⊢ (𝜑 → 𝑄 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑄𝐷(𝐹‘𝑘)) ≤ 𝑅) ⇒ ⊢ (𝜑 → (𝑄𝐷𝑃) ≤ 𝑅) | ||
Theorem | nglmle 23908* | 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) & ⊢ (𝜑 → 𝐺 ∈ NrmGrp) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) & ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑁‘(𝐹‘𝑘)) ≤ 𝑅) ⇒ ⊢ (𝜑 → (𝑁‘𝑃) ≤ 𝑅) | ||
Theorem | lmclim 23909 | 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 = (ℤ≥‘𝑀) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝑍 ⊆ dom 𝐹) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐹 ⇝ 𝑃))) | ||
Theorem | lmclimf 23910 | 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 = (ℤ≥‘𝑀) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) | ||
Theorem | metelcls 23911* | 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 9860. 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) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃))) | ||
Theorem | metcld 23912* | 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) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ∀𝑥∀𝑓((𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ 𝑆))) | ||
Theorem | metcld2 23913 | 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) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((⇝𝑡‘𝐽) “ (𝑆 ↑m ℕ)) ⊆ 𝑆)) | ||
Theorem | caubl 23914* | 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.) |
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ⟶(𝑋 × ℝ+)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))) & ⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑛 ∈ ℕ (2nd ‘(𝐹‘𝑛)) < 𝑟) ⇒ ⊢ (𝜑 → (1st ∘ 𝐹) ∈ (Cau‘𝐷)) | ||
Theorem | caublcls 23915* | 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.) |
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ⟶(𝑋 × ℝ+)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹‘𝐴)))) | ||
Theorem | metcnp4 23916* | 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) & ⊢ (𝜑 → 𝐶 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑌)) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ 𝑓(⇝𝑡‘𝐽)𝑃) → (𝐹 ∘ 𝑓)(⇝𝑡‘𝐾)(𝐹‘𝑃))))) | ||
Theorem | metcn4 23917* | 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) & ⊢ (𝜑 → 𝐶 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑌)) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑓(𝑓:ℕ⟶𝑋 → ∀𝑥(𝑓(⇝𝑡‘𝐽)𝑥 → (𝐹 ∘ 𝑓)(⇝𝑡‘𝐾)(𝐹‘𝑥))))) | ||
Theorem | iscmet3i 23918* | 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 (⇝𝑡‘𝐽)) ⇒ ⊢ 𝐷 ∈ (CMet‘𝑋) | ||
Theorem | lmcau 23919 | 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 (⇝𝑡‘𝐽) ⊆ (Cau‘𝐷)) | ||
Theorem | flimcfil 23920 | 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)) | ||
Theorem | metsscmetcld 23921 | A complete subspace of a metric space is closed in the parent space. Formerly part of proof for cmetss 23922. (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)) | ||
Theorem | cmetss 23922 | 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))) | ||
Theorem | equivcmet 23923* | If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 23906, metss2 23125, 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 ℝ induced by the usual homeomorphism from (0, 1) against the usual metric 𝐶 on ℝ and against the discrete metric 𝐸 on ℝ. 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.) |
⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → 𝑆 ∈ ℝ+) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ (𝑆 · (𝑥𝐶𝑦))) ⇒ ⊢ (𝜑 → (𝐶 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ (CMet‘𝑋))) | ||
Theorem | relcmpcmet 23924* | 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) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp) ⇒ ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) | ||
Theorem | cmpcmet 23925 | A compact metric space is complete. One half of heibor 35103. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐽 ∈ Comp) ⇒ ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) | ||
Theorem | cfilucfil3 23926 | 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))) | ||
Theorem | cfilucfil4 23927 | 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))) | ||
Theorem | cncmet 23928 | 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) | ||
Theorem | recmet 23929 | 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) | ||
Theorem | bcthlem1 23930* | Lemma for bcth 23935. Substitutions for the function 𝐹. (Contributed by Mario Carneiro, 9-Jan-2014.) |
⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {〈𝑥, 𝑟〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))}) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝐵 ∈ (𝑋 × ℝ+))) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ (𝐶 ∈ (𝑋 × ℝ+) ∧ (2nd ‘𝐶) < (1 / 𝐴) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘𝐶)) ⊆ (((ball‘𝐷)‘𝐵) ∖ (𝑀‘𝐴))))) | ||
Theorem | bcthlem2 23931* | Lemma for bcth 23935. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.) |
⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {〈𝑥, 𝑟〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))}) & ⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽)) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ+)) & ⊢ (𝜑 → (𝑔‘1) = 〈𝐶, 𝑅〉) & ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))) ⇒ ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))) | ||
Theorem | bcthlem3 23932* | Lemma for bcth 23935. 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) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {〈𝑥, 𝑟〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))}) & ⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽)) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ+)) & ⊢ (𝜑 → (𝑔‘1) = 〈𝐶, 𝑅〉) & ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))) ⇒ ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝐴 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘𝐴))) | ||
Theorem | bcthlem4 23933* | Lemma for bcth 23935. 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. (𝑈 ∖ 𝐴) ≠ ∅. (Contributed by Mario Carneiro, 7-Jan-2014.) |
⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {〈𝑥, 𝑟〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))}) & ⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽)) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ+)) & ⊢ (𝜑 → (𝑔‘1) = 〈𝐶, 𝑅〉) & ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))) ⇒ ⊢ (𝜑 → ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀) ≠ ∅) | ||
Theorem | bcthlem5 23934* |
Lemma for bcth 23935. The proof makes essential use of the Axiom
of
Dependent Choice axdc4uz 13355, 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, 𝑧 ∖ 𝑀(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 23931) 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 23933). 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 23932) 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) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {〈𝑥, 𝑟〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))}) & ⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽)) & ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅) ⇒ ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) = ∅) | ||
Theorem | bcth 23935* | 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 𝑀‘𝑘 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 23934 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)‘(𝑀‘𝑘)) ≠ ∅) | ||
Theorem | bcth2 23936* | 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)‘(𝑀‘𝑘)) ≠ ∅) | ||
Theorem | bcth3 23937* | 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)‘(𝑀‘𝑘)) = 𝑋) → ((cls‘𝐽)‘∩ ran 𝑀) = 𝑋) | ||
Syntax | ccms 23938 | Extend class notation with the class of complete metric spaces. |
class CMetSp | ||
Syntax | cbn 23939 | Extend class notation with the class of Banach spaces. |
class Ban | ||
Syntax | chl 23940 | Extend class notation with the class of subcomplex Hilbert spaces. |
class ℂHil | ||
Definition | df-cms 23941* | 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)} | ||
Definition | df-bn 23942 | 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} | ||
Definition | df-hl 23943 | 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) | ||
Theorem | isbn 23944 | 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)) | ||
Theorem | bnsca 23945 | 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) | ||
Theorem | bnnvc 23946 | A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec) | ||
Theorem | bnnlm 23947 | A Banach space is a normed module. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmMod) | ||
Theorem | bnngp 23948 | A Banach space is a normed group. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmGrp) | ||
Theorem | bnlmod 23949 | A Banach space is a left module. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ (𝑊 ∈ Ban → 𝑊 ∈ LMod) | ||
Theorem | bncms 23950 | A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp) | ||
Theorem | iscms 23951 | 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))) | ||
Theorem | cmscmet 23952 | 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)) | ||
Theorem | bncmet 23953 | 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)) | ||
Theorem | cmsms 23954 | A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ (𝐺 ∈ CMetSp → 𝐺 ∈ MetSp) | ||
Theorem | cmspropd 23955 | Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵))) & ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) ⇒ ⊢ (𝜑 → (𝐾 ∈ CMetSp ↔ 𝐿 ∈ CMetSp)) | ||
Theorem | cmssmscld 23956 | The restriction of a metric space is closed if it is complete. (Contributed by AV, 9-Oct-2022.) |
⊢ 𝐾 = (𝑀 ↾s 𝐴) & ⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐽 = (TopOpen‘𝑀) ⇒ ⊢ ((𝑀 ∈ MetSp ∧ 𝐴 ⊆ 𝑋 ∧ 𝐾 ∈ CMetSp) → 𝐴 ∈ (Clsd‘𝐽)) | ||
Theorem | cmsss 23957 | The restriction of a complete metric space is complete iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝐾 = (𝑀 ↾s 𝐴) & ⊢ 𝑋 = (Base‘𝑀) & ⊢ 𝐽 = (TopOpen‘𝑀) ⇒ ⊢ ((𝑀 ∈ CMetSp ∧ 𝐴 ⊆ 𝑋) → (𝐾 ∈ CMetSp ↔ 𝐴 ∈ (Clsd‘𝐽))) | ||
Theorem | lssbn 23958 | A subspace of a Banach space is a Banach space iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) ⇒ ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ Ban ↔ 𝑈 ∈ (Clsd‘𝐽))) | ||
Theorem | cmetcusp1 23959 | 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) | ||
Theorem | cmetcusp 23960 | 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) | ||
Theorem | cncms 23961 | The field of complex numbers is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ ℂfld ∈ CMetSp | ||
Theorem | cnflduss 23962 | 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 ∘ − )) | ||
Theorem | cnfldcusp 23963 | The field of complex numbers is a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.) |
⊢ ℂfld ∈ CUnifSp | ||
Theorem | resscdrg 23964 | The real numbers are a subset of any complete subfield in the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝐹 = (ℂfld ↾s 𝐾) ⇒ ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → ℝ ⊆ 𝐾) | ||
Theorem | cncdrg 23965 | The only complete subfields of the complex numbers are ℝ and ℂ. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝐹 = (ℂfld ↾s 𝐾) ⇒ ⊢ ((𝐾 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ DivRing ∧ 𝐹 ∈ CMetSp) → 𝐾 ∈ {ℝ, ℂ}) | ||
Theorem | srabn 23966 | 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) ∧ (𝑊 ↾s 𝑆) ∈ DivRing))) | ||
Theorem | rlmbn 23967 | 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) | ||
Theorem | ishl 23968 | 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)) | ||
Theorem | hlbn 23969 | Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) |
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban) | ||
Theorem | hlcph 23970 | Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil) | ||
Theorem | hlphl 23971 | 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) | ||
Theorem | hlcms 23972 | Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.) |
⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp) | ||
Theorem | hlprlem 23973 | Lemma for hlpr 23975. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ ℂHil → (𝐾 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝐾) ∈ DivRing ∧ (ℂfld ↾s 𝐾) ∈ CMetSp)) | ||
Theorem | hlress 23974 | The scalar field of a subcomplex Hilbert space contains ℝ. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ ℂHil → ℝ ⊆ 𝐾) | ||
Theorem | hlpr 23975 | The scalar field of a subcomplex Hilbert space is either ℝ or ℂ. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ ℂHil → 𝐾 ∈ {ℝ, ℂ}) | ||
Theorem | ishl2 23976 | A Hilbert space is a complete subcomplex pre-Hilbert space over ℝ or ℂ. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil ∧ 𝐾 ∈ {ℝ, ℂ})) | ||
Theorem | cphssphl 23977 | 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 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ Ban) → 𝑋 ∈ ℂHil) | ||
Theorem | cmslssbn 23978 | 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 23942. (Contributed by AV, 8-Oct-2022.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp) ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → 𝑋 ∈ Ban) | ||
Theorem | cmscsscms 23979 | 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 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (ClSubSp‘𝑊) ⇒ ⊢ (((𝑊 ∈ CMetSp ∧ 𝑊 ∈ ℂPreHil) ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ CMetSp) | ||
Theorem | bncssbn 23980 | 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 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (ClSubSp‘𝑊) ⇒ ⊢ (((𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil) ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ Ban) | ||
Theorem | cssbn 23981 | 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 Cℋ (df-ch 29001) of closed subspaces of a Hilbert space. It may be superseded by cmslssbn 23978. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 6-Oct-2022.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐷 = ((dist‘𝑊) ↾ (𝑈 × 𝑈)) ⇒ ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ Ban) | ||
Theorem | csschl 23982 | 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 Cℋ (df-ch 29001) 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 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝐷 = ((dist‘𝑊) ↾ (𝑈 × 𝑈)) & ⊢ (Scalar‘𝑊) = ℂfld ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (𝑋 ∈ ℂHil ∧ (Scalar‘𝑋) = ℂfld)) | ||
Theorem | cmslsschl 23983 | A complete linear subspace of a subcomplex Hilbert space is a subcomplex Hilbert space. (Contributed by AV, 8-Oct-2022.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂHil ∧ 𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ ℂHil) | ||
Theorem | chlcsschl 23984 | 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 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (ClSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ ℂHil) | ||
Theorem | retopn 23985 | The topology of the real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ (topGen‘ran (,)) = (TopOpen‘ℝfld) | ||
Theorem | recms 23986 | The real numbers form a complete metric space. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
⊢ ℝfld ∈ CMetSp | ||
Theorem | reust 23987 | The Uniform structure of the real numbers. (Contributed by Thierry Arnoux, 14-Feb-2018.) |
⊢ (UnifSt‘ℝfld) = (metUnif‘((dist‘ℝfld) ↾ (ℝ × ℝ))) | ||
Theorem | recusp 23988 | The real numbers form a complete uniform space. (Contributed by Thierry Arnoux, 17-Dec-2017.) |
⊢ ℝfld ∈ CUnifSp | ||
Syntax | crrx 23989 | Extend class notation with generalized real Euclidean spaces. |
class ℝ^ | ||
Syntax | cehl 23990 | Extend class notation with real Euclidean spaces. |
class 𝔼hil | ||
Definition | df-rrx 23991 | Define the function associating with a set the free real vector space on that set, equipped with the natural inner product and norm. 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^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖))) | ||
Definition | df-ehl 23992 | Define a function generating the real Euclidean spaces of finite dimension. The case 𝑛 = 0 corresponds to a space of dimension 0, that is, limited to a neutral element (see ehl0 24023). Members of this family of spaces are Hilbert spaces, as shown in - ehlhl . (Contributed by Thierry Arnoux, 16-Jun-2019.) |
⊢ 𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛))) | ||
Theorem | rrxval 23993 | Value of the generalized Euclidean space. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
⊢ 𝐻 = (ℝ^‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) | ||
Theorem | rrxbase 23994* | 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.) |
⊢ 𝐻 = (ℝ^‘𝐼) & ⊢ 𝐵 = (Base‘𝐻) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {𝑓 ∈ (ℝ ↑m 𝐼) ∣ 𝑓 finSupp 0}) | ||
Theorem | rrxprds 23995 | Expand the definition of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
⊢ 𝐻 = (ℝ^‘𝐼) & ⊢ 𝐵 = (Base‘𝐻) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s 𝐵))) | ||
Theorem | rrxip 23996* | The inner product of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
⊢ 𝐻 = (ℝ^‘𝐼) & ⊢ 𝐵 = (Base‘𝐻) ⇒ ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (ℝfld Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))) = (·𝑖‘𝐻)) | ||
Theorem | rrxnm 23997* | The norm of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
⊢ 𝐻 = (ℝ^‘𝐼) & ⊢ 𝐵 = (Base‘𝐻) ⇒ ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝐵 ↦ (√‘(ℝfld Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))))) = (norm‘𝐻)) | ||
Theorem | rrxcph 23998 | Generalized Euclidean real spaces are subcomplex pre-Hilbert spaces. (Contributed by Thierry Arnoux, 23-Jun-2019.) (Proof shortened by AV, 22-Jul-2019.) |
⊢ 𝐻 = (ℝ^‘𝐼) & ⊢ 𝐵 = (Base‘𝐻) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐻 ∈ ℂPreHil) | ||
Theorem | rrxds 23999* | The distance over generalized Euclidean spaces. Compare with df-rrn 35108. (Contributed by Thierry Arnoux, 20-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.) |
⊢ 𝐻 = (ℝ^‘𝐼) & ⊢ 𝐵 = (Base‘𝐻) ⇒ ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (√‘(ℝfld Σg (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (dist‘𝐻)) | ||
Theorem | rrxvsca 24000 | The scalar product over generalized Euclidean spaces is the componentwise real number multiplication. (Contributed by Thierry Arnoux, 18-Jan-2023.) |
⊢ 𝐻 = (ℝ^‘𝐼) & ⊢ 𝐵 = (Base‘𝐻) & ⊢ ∙ = ( ·𝑠 ‘𝐻) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐻)) ⇒ ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) |
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