Theorems · Theorem · harmonic analysis

BoundedContinuousFunction.ext_of_char_eq

∀ {V : Type u_1} {W : Type u_2} [inst : AddCommGroup V] [inst_1 : Module ℝ V] [inst_2 : TopologicalSpace V]
  [inst_3 : AddCommGroup W] [inst_4 : Module ℝ W] [inst_5 : TopologicalSpace W] {e : AddChar ℝ Circle}
  {L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ} (he : Continuous ⇑e),
  e ≠ 1 →
    ∀ (hL : Continuous fun p => (L p.1) p.2),
      (∀ (v : V), v ≠ 0 → L v ≠ 0) →
        ∀ {v v' : V},
          (∀ (w : W), (BoundedContinuousFunction.char he hL w) v = (BoundedContinuousFunction.char he hL w) v') → v = v'

If e and L are non-trivial, then char he hL w, w : W separates points in V.

Defined in

Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar

Cited by

1 results in Mathlib

Foundations

Depth 149 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupModuleTopologicalSpaceAddCommGroupModuleTopologicalSpace