Theorems · Theorem · functional analysis

tendsto_iff_forall_eval_tendsto_topDualPairing

∀ {α : Type u_1} {𝕜 : Type u_2} {E : Type u_4} [inst : CommSemiring 𝕜] [inst_1 : TopologicalSpace 𝕜]
  [inst_2 : ContinuousAdd 𝕜] [inst_3 : ContinuousConstSMul 𝕜 𝕜] [inst_4 : AddCommMonoid E] [inst_5 : Module 𝕜 E]
  [inst_6 : TopologicalSpace E] {l : Filter α} {f : α → WeakDual 𝕜 E} {x : WeakDual 𝕜 E},
  Filter.Tendsto f l (nhds x) ↔
    ∀ (y : E), Filter.Tendsto (fun i => ((topDualPairing 𝕜 E) (f i)) y) l (nhds (((topDualPairing 𝕜 E) x) y))

Defined in

Mathlib.Topology.Algebra.Module.Spaces.WeakDual

Cited by

1 results in Mathlib

Foundations

Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringTopologicalSpaceContinuousAddContinuousConstSMulAddCommMonoidModuleTopologicalSpace