Theorems · Theorem · functional analysis

withSeminorms_iff_mem_nhds_isVonNBounded

∀ {𝕜 : Type u_2} {E : Type u_6} [inst : NontriviallyNormedField 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
  [inst_3 : TopologicalSpace E] [IsTopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {p : Seminorm 𝕜 E},
  (WithSeminorms fun x => p) ↔ p.ball 0 1 ∈ nhds 0 ∧ Bornology.IsVonNBounded 𝕜 (p.ball 0 1)

In a topological vector space, the topology is generated by a single seminorm p iff the unit ball for this seminorm is a bounded neighborhood of 0.

Defined in

Mathlib.Analysis.LocallyConvex.WithSeminorms

Cited by

1 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupContinuousConstSMul