Theorems · Theorem · functional analysis

LinearMap.eq_adjoint_iff_basis_left

∀ {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] {ι : Type u_5} (b : Module.Basis ι 𝕜 E)
  (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E),
  A = LinearMap.adjoint B ↔ ∀ (i : ι) (y : F), inner 𝕜 (A (b i)) y = inner 𝕜 (b i) (B y)

Defined in

Mathlib.Analysis.InnerProductSpace.Adjoint

Cited by

0 results in Mathlib

Foundations

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

Assumes

RCLikeNormedAddCommGroupNormedAddCommGroupInnerProductSpaceInnerProductSpaceFiniteDimensionalFiniteDimensional