Theorems · Theorem · functional analysis

WithSeminorms.equicontinuous_TFAE

∀ {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] [hu : IsUniformAddGroup F],
  WithSeminorms q →
    ∀ [ContinuousSMul 𝕜 E] (f : κ → E →ₛₗ[σ₁₂] F),
      [EquicontinuousAt (DFunLike.coe ∘ f) 0, Equicontinuous (DFunLike.coe ∘ f),
          UniformEquicontinuous (DFunLike.coe ∘ f), ∀ (i : ι'), ∃ p, Continuous ⇑p ∧ ∀ (k : κ), (q i).comp (f k) ≤ p,
          ∀ (i : ι'), BddAbove (Set.range fun k => (q i).comp (f k)) ∧ Continuous (⨆ k, ⇑((q i).comp (f k)))].TFAE

Let E and F be two topological vector spaces over a NontriviallyNormedField, and assume that the topology of F is generated by some family of seminorms q. For a family f of linear maps from E to F, the following are equivalent: * f is equicontinuous at 0. * f is equicontinuous. * f is uniformly equicontinuous. * For each q i, the family of seminorms k ↦ (q i) ∘ (f k) is bounded by some continuous seminorm p on E. * For each q i, the seminorm ⊔ k, (q i) ∘ (f k) is well-defined and continuous. In particular, if you can determine all continuous seminorms on E, that gives you a complete characterization of equicontinuity for linear maps from E to F. For example E and F are both normed spaces, you get NormedSpace.equicontinuous_TFAE.

Defined in

Mathlib.Analysis.LocallyConvex.WithSeminorms

Cited by

2 results in Mathlib

Foundations

Depth 163 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupModuleNormedFieldAddCommGroupModuleRingHomIsometricUniformSpaceIsUniformAddGroupUniformSpaceIsUniformAddGroupContinuousSMul