Theorems · Definition · functional analysis

Unitary.linearIsometryEquiv

{𝕜 : Type u_1} →
  [inst : RCLike 𝕜] →
    {H : Type u_5} →
      [inst_1 : NormedAddCommGroup H] →
        [inst_2 : InnerProductSpace 𝕜 H] → [inst_3 : CompleteSpace H] → ↥(unitary (H →L[𝕜] H)) ≃* (H ≃ₗᵢ[𝕜] H)

The unitary elements of continuous linear maps on a Hilbert space coincide with the linear isometric equivalences on that Hilbert space.

Defined in

Mathlib.Analysis.InnerProductSpace.Adjoint

Cited by

4 results in Mathlib

Foundations

Depth 199 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpace