Theorems · Definition · functional analysis

AffineEquiv.toContinuousAffineEquiv

{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] → (PE ≃ᵃ[𝕜] PF) ≃ (PE ≃ᴬ[𝕜] PF)

Reinterpret an affine equivalence as a continuous affine equivalence in finite dimension.

Defined in

Mathlib.Analysis.Normed.Module.FiniteDimension

Cited by

6 results in Mathlib

Foundations

Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceMetricSpaceNormedAddTorsorMetricSpaceNormedAddTorsorFiniteDimensional