Theorems · Definition · functional analysis

IsometryEquiv.toRealAffineIsometryEquiv

{E : Type u_1} →
  {PE : Type u_2} →
    {F : Type u_3} →
      {PF : Type u_4} →
        [inst : NormedAddCommGroup E] →
          [inst_1 : NormedSpace ℝ E] →
            [inst_2 : MetricSpace PE] →
              [inst_3 : NormedAddTorsor E PE] →
                [inst_4 : NormedAddCommGroup F] →
                  [inst_5 : NormedSpace ℝ F] →
                    [inst_6 : MetricSpace PF] → [inst_7 : NormedAddTorsor F PF] → PE ≃ᵢ PF → PE ≃ᵃⁱ[ℝ] PF

Mazur-Ulam Theorem: if f is an isometric bijection between two normed add-torsors over normed vector spaces over ℝ, then f is an affine isometry equivalence.

Defined in

Mathlib.Analysis.Normed.Affine.MazurUlam

Cited by

2 results in Mathlib

Foundations

Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMetricSpaceNormedAddTorsorNormedAddCommGroupNormedSpaceMetricSpaceNormedAddTorsor