Theorems · Theorem · functional analysis

FiniteDimensional.of_totallyBounded_nhds

∀ (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] {x : Eᵤ}
  {U : Set Eᵤ}, U ∈ nhds x → TotallyBounded U → FiniteDimensional 𝕜 Eᵤ

Riesz's theorem: if a T2 topological vector space over a complete non-trivial normed field admits a totally bounded neighborhood of some point, then it is finite-dimensional.

Defined in

Mathlib.Topology.Algebra.Module.FiniteDimension

Cited by

1 results in Mathlib

Foundations

Depth 175 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldCompleteSpaceAddCommGroupModuleUniformSpaceT2SpaceIsUniformAddGroupContinuousSMul