Theorems · Definition · functional analysis

LinearEquiv.isometryOfInner

{𝕜 : Type u_1} →
  {E : Type u_2} →
    [inst : RCLike 𝕜] →
      [inst_1 : SeminormedAddCommGroup E] →
        [inst_2 : InnerProductSpace 𝕜 E] →
          {E' : Type u_7} →
            [inst_3 : SeminormedAddCommGroup E'] →
              [inst_4 : InnerProductSpace 𝕜 E'] →
                (f : E ≃ₗ[𝕜] E') → (∀ (x y : E), inner 𝕜 (f x) (f y) = inner 𝕜 x y) → E ≃ₗᵢ[𝕜] E'

A linear equivalence that preserves the inner product is a linear isometric equivalence.

Defined in

Mathlib.Analysis.InnerProductSpace.LinearMap

Cited by

3 results in Mathlib

Foundations

Depth 157 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeSeminormedAddCommGroupInnerProductSpaceSeminormedAddCommGroupInnerProductSpace