Uniqueness principle for analytic functions

We show that two analytic functions which coincide around a point coincide on whole connected sets, in AnalyticOn.eqOn_of_preconnected_of_eventuallyEq.

theorem AnalyticOn.eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {U : Set E} (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : (nhds z₀).EventuallyEq f 0) : Set.EqOn f 0 U

If an analytic function vanishes around a point, then it is uniformly zero along a connected set. Superseded by eqOn_zero_of_preconnected_of_locally_zero which does not assume completeness of the target space.

theorem AnalyticOn.eqOn_zero_of_preconnected_of_eventuallyEq_zero {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {U : Set E} (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : (nhds z₀).EventuallyEq f 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 eqOn_zero_of_preconnected_of_frequently_eq_zero.

theorem AnalyticOn.eqOn_of_preconnected_of_eventuallyEq {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {g : E → F} {U : Set E} (hf : AnalyticOn 𝕜 f U) (hg : AnalyticOn 𝕜 g U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfg : (nhds z₀).EventuallyEq f g) : Set.EqOn f g U

The identity principle for analytic functions: If two analytic functions coincide in a whole neighborhood of a point z₀, then they coincide globally along a connected set. For a one-dimensional version assuming only that the functions coincide at some points arbitrarily close to z₀, see eqOn_of_preconnected_of_frequently_eq.

theorem AnalyticOn.eq_of_eventuallyEq {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {g : E → F} [PreconnectedSpace E] (hf : AnalyticOn 𝕜 f Set.univ) (hg : AnalyticOn 𝕜 g Set.univ) {z₀ : E} (hfg : (nhds z₀).EventuallyEq f g) : f = g

The identity principle for analytic functions: If two analytic functions on a normed space coincide in a neighborhood of a point z₀, then they coincide everywhere. For a one-dimensional version assuming only that the functions coincide at some points arbitrarily close to z₀, see eq_of_frequently_eq.