Theorems · Theorem · functional analysis

SeminormFamily.withSeminorms_iff_topologicalSpace_eq_iInf

∀ {𝕜 : Type u_2} {E : Type u_6} {ι : Type u_9} [inst : NormedField 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
  [t : TopologicalSpace E] [IsTopologicalAddGroup E] (p : SeminormFamily 𝕜 E ι),
  WithSeminorms p ↔ t = ⨅ i, PseudoMetricSpace.toUniformSpace.toTopologicalSpace

The topology induced by a family of seminorms is exactly the infimum of the ones induced by each seminorm individually. We express this as a characterization of WithSeminorms p.

Defined in

Mathlib.Analysis.LocallyConvex.WithSeminorms

Cited by

3 results in Mathlib

Foundations

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

Assumes

NormedFieldAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroup