Theorems · Theorem · several complex variables

MDifferentiableOn.eqOn_of_isPreconnected_of_isMaxOn_norm

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {F : Type u_2} [inst_2 : NormedAddCommGroup F]
  [inst_3 : NormedSpace ℂ F] {H : Type u_3} [inst_4 : TopologicalSpace H] {I : ModelWithCorners ℂ E H} [I.Boundaryless]
  {M : Type u_4} [inst_6 : TopologicalSpace M] [inst_7 : ChartedSpace H M] [IsManifold I 1 M] [StrictConvexSpace ℝ F]
  {f : M → F} {U : Set M} {c : M},
  MDiff[U] f → IsPreconnected U → IsOpen U → c ∈ U → IsMaxOn (norm ∘ f) U c → Set.EqOn f (Function.const M (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.Geometry.Manifold.Complex

Cited by

0 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceModelWithCorners.BoundarylessTopologicalSpaceChartedSpaceIsManifoldStrictConvexSpace