Theorems · Theorem · linear algebra
LinearMap.finrank_range_dualMap_eq_finrank_range
∀ {K : Type u_1} {V₁ : Type u_2} {V₂ : Type u_3} [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),
Module.finrank K ↥f.dualMap.range = Module.finrank K ↥f.range- Defined in
- Mathlib.LinearAlgebra.Dual.Lemmas
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites24
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement and proof · cited by 10,215
- Fieldstatement and proof · cited by 7,404
- Submodulestatement and proof · cited by 7,192
- Module.finrankstatement and proof · cited by 1,770
- LinearMap.compproof · cited by 1,642
- LinearMap.rangestatement and proof · cited by 893
- Submodule.mapproof · cited by 614
- Module.Dualstatement and proof · cited by 583
- Submodule.subtypeproof · cited by 480
Cited by1
Results whose statement or proof uses this declaration.
- Matrix.rank_transposeproof · cited by 3