Theorems · Theorem · complex analysis

Complex.eqOn_of_isPreconnected_of_isMaxOn_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} {U : Set E} {c : E},
  IsPreconnected U →
    IsOpen U → DifferentiableOn ℂ f U → c ∈ U → IsMaxOn (norm ∘ f) U c → Set.EqOn f (Function.const E (f c)) U

Maximum modulus principle on a connected set. Let U be a (pre)connected open set in a complex normed space. Let f : E → F be a function that is complex differentiable on U. Suppose that ‖f x‖ takes its maximum value on U at c ∈ U. Then f x = f c for all x ∈ U. TODO: change assumption from IsMaxOn to IsLocalMax.

Defined in

Mathlib.Analysis.Complex.AbsMax

Cited by

2 results in Mathlib

Foundations

Depth 291 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceStrictConvexSpace