Theorems · Theorem · functional analysis

SeminormFamily.withSeminorms_iff_uniformSpace_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]
  [u : UniformSpace E] [IsUniformAddGroup E] (p : SeminormFamily 𝕜 E ι),
  WithSeminorms p ↔ u = ⨅ i, PseudoMetricSpace.toUniformSpace

The uniform structure 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

1 results in Mathlib

Foundations

Depth 162 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedFieldAddCommGroupModuleUniformSpaceIsUniformAddGroup