Theorems · Theorem · functional analysis

WithSeminorms.separating_of_T1

∀ {𝕜 : Type u_2} {E : Type u_6} {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} [T1Space E],
  WithSeminorms p → ∀ (x : E), x ≠ 0 → ∃ i, (p i) x ≠ 0

A family of seminorms inducing a T₁ topology is separating.

Defined in

Mathlib.Analysis.LocallyConvex.WithSeminorms

Cited by

1 results in Mathlib

Foundations

Depth 136 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedFieldAddCommGroupModuleTopologicalSpaceT1Space