Theorems · Theorem · complex analysis
Complex.eventually_eq_of_isLocalMax_norm
∀ {E : Type u} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {F : Type v} [inst_2 : NormedAddCommGroup F]
[inst_3 : NormedSpace ℂ F] [StrictConvexSpace ℝ F] {f : E → F} {c : E},
(∀ᶠ (z : E) in nhds c, DifferentiableAt ℂ f z) → IsLocalMax (norm ∘ f) c → ∀ᶠ (y : E) in nhds c, f y = f cMaximum modulus principle: if f : E → F is complex differentiable in a neighborhood of c
and the norm ‖f z‖ has a local maximum at c, then f is locally constant in a neighborhood
of c.
- Defined in
- Mathlib.Analysis.Complex.AbsMax
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 291 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- Complexstatement and proof · cited by 5,565
- nhdsstatement and proof · cited by 5,554
- Norm.normstatement and proof · cited by 5,413
- Filter.Eventuallystatement and proof · cited by 3,134
- closureproof · cited by 1,254
- Metric.ballproof · cited by 735
- Metric.closedBallproof · cited by 704
- DifferentiableAtstatement and proof · cited by 617
- Filter.Eventually.andproof · cited by 157
Cited by1
Results whose statement or proof uses this declaration.
- Complex.eventually_eq_or_eq_zero_of_isLocalMin_normproof · cited by 1