Theorems · Theorem · linear algebra

LinearMap.dualMap_surjective_iff

∀ {K : Type u_3} {V₁ : Type u_4} {V₂ : Type u_5} [inst : Field K] [inst_1 : AddCommGroup V₁] [inst_2 : Module K V₁]
  [inst_3 : AddCommGroup V₂] [inst_4 : Module K V₂] {f : V₁ →ₗ[K] V₂},
  Function.Surjective ⇑f.dualMap ↔ Function.Injective ⇑f

For vector spaces, f.dualMap is surjective if and only if f is injective

Defined in

Mathlib.LinearAlgebra.Dual.Lemmas

Cited by

2 results in Mathlib

Foundations

Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldAddCommGroupModuleAddCommGroupModule