Theorems · Theorem · linear algebra

LinearMap.nondegenerate_restrict_of_disjoint_orthogonal

∀ {R : Type u_1} {M : Type u_4} {M₁ : Type u_5} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [inst_3 : AddCommGroup M₁] [inst_4 : Module R M₁] {B : M →ₗ[R] M →ₗ[R] M₁},
  B.IsRefl → ∀ {W : Submodule R M}, Disjoint W (Submodule.orthogonalBilin B W) → (B.domRestrict₁₂ W W).Nondegenerate

The restriction of a reflexive bilinear map B onto a submodule W is nondegenerate if W has trivial intersection with its orthogonal complement, that is Disjoint W (W.orthogonalBilin B).

Defined in

Mathlib.LinearAlgebra.SesquilinearForm.Orthogonal

Cited by

1 results in Mathlib

Foundations

Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModule