Theorems · Theorem · functional analysis

AffineMap.antilipschitzWith_of_finiteDimensional

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type v} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type w} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] [CompleteSpace 𝕜]
  {PE : Type u_1} {PF : Type u_2} [inst_6 : MetricSpace PE] [inst_7 : NormedAddTorsor E PE] [inst_8 : MetricSpace PF]
  [inst_9 : NormedAddTorsor F PF] [FiniteDimensional 𝕜 E] {f : PE →ᵃ[𝕜] PF},
  Function.Injective ⇑f → ∃ K, AntilipschitzWith K ⇑f

An injective affine map from a finite-dimensional space is automatically anti-Lipschitz.

Defined in

Mathlib.Analysis.Normed.Module.FiniteDimension

Cited by

0 results in Mathlib

Foundations

Depth 178 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceMetricSpaceNormedAddTorsorMetricSpaceNormedAddTorsorFiniteDimensional