Theorems · Theorem · functional analysis

LinearMap.self_comp_adjoint_injective_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),
  Function.Injective (⇑A ∘ ⇑(LinearMap.adjoint A)) ↔ Function.Injective ⇑(LinearMap.adjoint A)

This lemma uses the simp-normal form ⇑A ∘ ⇑(A.adjoint) instead of ⇑(A ∘ₗ A.adjoint) (note the difference between ∘ and ∘ₗ). You may need to rewrite with LinearMap.coe_comp before applying this lemma.

Defined in

Mathlib.Analysis.InnerProductSpace.Adjoint

Cited by

0 results in Mathlib

Foundations

Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedAddCommGroupInnerProductSpaceInnerProductSpaceFiniteDimensionalFiniteDimensional