Theorems · Definition · operator theory

LinearPMap.IsFormalAdjoint

{𝕜 : Type u_1} →
  {E : Type u_2} →
    {F : Type u_3} →
      [inst : RCLike 𝕜] →
        [inst_1 : NormedAddCommGroup E] →
          [inst_2 : InnerProductSpace 𝕜 E] →
            [inst_3 : NormedAddCommGroup F] → [inst_4 : InnerProductSpace 𝕜 F] → (E →ₗ.[𝕜] F) → (F →ₗ.[𝕜] E) → Prop

An operator T is a formal adjoint of S if for all x in the domain of T and y in the domain of S, we have that ⟪T x, y⟫ = ⟪x, S y⟫.

Defined in

Mathlib.Analysis.InnerProductSpace.LinearPMap

Cited by

3 results in Mathlib

Foundations

Depth 46 from the axioms · uses propext, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpace