Theorems · Theorem · functional analysis

FiniteDimensional.of_totallyBounded_nhds_zero

∀ (𝕜 : Type u_4) [inst : NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {Eᵤ : Type u_6} [inst_2 : AddCommGroup Eᵤ]
  [inst_3 : Module 𝕜 Eᵤ] [inst_4 : UniformSpace Eᵤ] [T2Space Eᵤ] [IsUniformAddGroup Eᵤ] [ContinuousSMul 𝕜 Eᵤ]
  {U : Set Eᵤ}, U ∈ nhds 0 → TotallyBounded U → FiniteDimensional 𝕜 Eᵤ

Riesz's theorem: a T2 topological vector space over a complete non-trivial normed field which admits a totally bounded neighborhood of 0 is finite-dimensional.

Defined in

Mathlib.Topology.Algebra.Module.FiniteDimension

Cited by

3 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldCompleteSpaceAddCommGroupModuleUniformSpaceT2SpaceIsUniformAddGroupContinuousSMul