Theorems · Theorem · functional analysis

LinearMap.eq_adjoint_iff

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedAddCommGroup F] [inst_3 : InnerProductSpace 𝕜 E] [inst_4 : InnerProductSpace 𝕜 F]
  [inst_5 : FiniteDimensional 𝕜 E] [inst_6 : FiniteDimensional 𝕜 F] (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E),
  A = LinearMap.adjoint B ↔ ∀ (x : E) (y : F), inner 𝕜 (A x) y = inner 𝕜 x (B y)

The adjoint is unique: a map A is the adjoint of B iff it satisfies ⟪A x, y⟫ = ⟪x, B y⟫ for all x and y.

Defined in

Mathlib.Analysis.InnerProductSpace.Adjoint

Cited by

2 results in Mathlib

Foundations

Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedAddCommGroupInnerProductSpaceInnerProductSpaceFiniteDimensionalFiniteDimensional