Theorems · Theorem · functional analysis

WithSeminorms.uniformEquicontinuous_iff_exists_continuous_seminorm

∀ {𝕜 : Type u_2} {𝕜₂ : Type u_3} {E : Type u_6} {F : Type u_7} {ι' : Type u_10} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E] [inst_3 : NormedField 𝕜₂] [inst_4 : AddCommGroup F]
  [inst_5 : Module 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [inst_6 : RingHomIsometric σ₁₂] {κ : Type u_11} {q : SeminormFamily 𝕜₂ F ι'}
  [inst_7 : UniformSpace E] [IsUniformAddGroup E] [u : UniformSpace F] [IsUniformAddGroup F],
  WithSeminorms q →
    ∀ [ContinuousSMul 𝕜 E] (f : κ → E →ₛₗ[σ₁₂] F),
      UniformEquicontinuous (DFunLike.coe ∘ f) ↔ ∀ (i : ι'), ∃ p, Continuous ⇑p ∧ ∀ (k : κ), (q i).comp (f k) ≤ p

Defined in

Mathlib.Analysis.LocallyConvex.WithSeminorms

Cited by

1 results in Mathlib

Foundations

Depth 164 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupModuleNormedFieldAddCommGroupModuleRingHomIsometricUniformSpaceIsUniformAddGroupUniformSpaceIsUniformAddGroupContinuousSMul