Theorems · Definition · functional analysis

LinearEquiv.isometryOfOrthonormal

{𝕜 : Type u_1} →
  {E : Type u_2} →
    [inst : RCLike 𝕜] →
      [inst_1 : SeminormedAddCommGroup E] →
        [inst_2 : InnerProductSpace 𝕜 E] →
          {ι : Type u_4} →
            {E' : Type u_7} →
              [inst_3 : SeminormedAddCommGroup E'] →
                [inst_4 : InnerProductSpace 𝕜 E'] →
                  (f : E ≃ₗ[𝕜] E') → {v : Module.Basis ι 𝕜 E} → Orthonormal 𝕜 ⇑v → Orthonormal 𝕜 (⇑f ∘ ⇑v) → E ≃ₗᵢ[𝕜] E'

A linear equivalence that sends an orthonormal basis to orthonormal vectors is a linear isometric equivalence.

Defined in

Mathlib.Analysis.InnerProductSpace.Orthonormal

Cited by

2 results in Mathlib

Foundations

Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeSeminormedAddCommGroupInnerProductSpaceSeminormedAddCommGroupInnerProductSpace