Theorems · Theorem · complex analysis

AnalyticOnNhd.eqOn_zero_of_preconnected_of_eventuallyEq_zero

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
  {U : Set E}, AnalyticOnNhd 𝕜 f U → IsPreconnected U → ∀ {z₀ : E}, z₀ ∈ U → f =ᶠ[nhds z₀] 0 → Set.EqOn f 0 U

The identity principle for analytic functions: If an analytic function vanishes in a whole neighborhood of a point z₀, then it is uniformly zero along a connected set. For a one-dimensional version assuming only that the function vanishes at some points arbitrarily close to z₀, see AnalyticOnNhd.eqOn_zero_of_preconnected_of_frequently_eq_zero.

Defined in

Mathlib.Analysis.Analytic.Uniqueness

Cited by

2 results in Mathlib

Foundations

Depth 188 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace