Theorems · Theorem · complex analysis
IsOpen.analyticOn_iff_analyticOnNhd
∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} {F : Type u_3} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F} {s : Set E},
IsOpen s → (AnalyticOn 𝕜 f s ↔ AnalyticOnNhd 𝕜 f s)On open sets, AnalyticOnNhd and AnalyticOn coincide
- Defined in
- Mathlib.Analysis.Analytic.Within
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites28
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Realproof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- ENNRealproof · cited by 9,879
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- IsOpenstatement and proof · cited by 2,400
- ENNReal.ofRealproof · cited by 863
- Metric.ballproof · cited by 735
- EDist.edistproof · cited by 735
- FormalMultilinearSeriesproof · cited by 615
- IsOpen.mem_nhdsproof · cited by 470
Cited by5
Results whose statement or proof uses this declaration.
- AnalyticOn.hasFPowerSeriesOnSubballproof · cited by 2
- ValueDistribution.logCounting_add_analyticOnproof · cited by 1
- Complex.analyticOn_iff_differentiableOnproof · cited by 1
- AnalyticOnNhd.fderiv_of_isOpenproof · cited by 1
- AnalyticOnNhd.iteratedFDeriv_of_isOpenproof · cited by 0