Theorems · Theorem · functional analysis

WithSeminorms.tendsto_nhds

∀ {𝕜 : Type u_2} {E : Type u_6} {F : Type u_7} {ι : Type u_9} [inst : NormedField 𝕜] [inst_1 : AddCommGroup E]
  [inst_2 : Module 𝕜 E] [inst_3 : TopologicalSpace E] {p : SeminormFamily 𝕜 E ι},
  WithSeminorms p →
    ∀ (u : F → E) {f : Filter F} (y₀ : E),
      Filter.Tendsto u f (nhds y₀) ↔ ∀ (i : ι) (ε : ℝ), 0 < ε → ∀ᶠ (x : F) in f, (p i) (u x - y₀) < ε

Convergence along filters for WithSeminorms.

Defined in

Mathlib.Analysis.LocallyConvex.WithSeminorms

Cited by

2 results in Mathlib

Foundations

Depth 165 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedFieldAddCommGroupModuleTopologicalSpace