Theorems · Theorem · functional analysis

UniformConvergenceCLM.hasBasis_nhds_zero_of_basis

∀ {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [inst : NormedField 𝕜₁] [inst_1 : NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3}
  (F : Type u_4) [inst_2 : AddCommGroup E] [inst_3 : Module 𝕜₁ E] [inst_4 : TopologicalSpace E]
  [inst_5 : AddCommGroup F] [inst_6 : Module 𝕜₂ F] [inst_7 : TopologicalSpace F] [inst_8 : IsTopologicalAddGroup F]
  {ι : Type u_6} (𝔖 : Set (Set E)),
  𝔖.Nonempty →
    DirectedOn (fun x1 x2 => x1 ⊆ x2) 𝔖 →
      ∀ {p : ι → Prop} {b : ι → Set F},
        (nhds 0).HasBasis p b → (nhds 0).HasBasis (fun Si => Si.1 ∈ 𝔖 ∧ p Si.2) fun Si => {f | ∀ x ∈ Si.1, f x ∈ b Si.2}

Defined in

Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM

Cited by

5 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedFieldNormedFieldAddCommGroupModuleTopologicalSpaceAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroup