Theorems · Theorem · linear algebra

LinearMap.injective_iff_forall_lt_finrank_singularValues_pos

∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  [inst_3 : FiniteDimensional 𝕜 E] {F : Type u_3} [inst_4 : NormedAddCommGroup F] [inst_5 : InnerProductSpace 𝕜 F]
  [inst_6 : FiniteDimensional 𝕜 F] (T : E →ₗ[𝕜] F),
  Function.Injective ⇑T ↔ ∀ i < Module.finrank 𝕜 E, 0 < T.singularValues i

7.68(a) from [axler2024]. Note that we have countably infinitely many singular values whereas there are only dim(domain(T)) singular values in [axler2024], so we modify the statement to account for this.

Defined in

Mathlib.Analysis.InnerProductSpace.SingularValues

Cited by

0 results in Mathlib

Foundations

Depth 261 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFiniteDimensionalNormedAddCommGroupInnerProductSpaceFiniteDimensional