Theorems · Theorem · linear algebra

LinearMap.separatingLeft_iff_ker_eq_bot

∀ {R : Type u_1} {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_5} {M₁ : Type u_6} {M₂ : Type u_7} [inst : CommSemiring R]
  [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [inst_3 : CommSemiring R₁] [inst_4 : AddCommMonoid M₁]
  [inst_5 : Module R₁ M₁] [inst_6 : CommSemiring R₂] [inst_7 : AddCommMonoid M₂] [inst_8 : Module R₂ M₂] {I₁ : R₁ →+* R}
  {I₂ : R₂ →+* R} {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M}, B.SeparatingLeft ↔ B.ker = ⊥

A bilinear map is left-separating if and only if it has a trivial kernel.

Defined in

Mathlib.LinearAlgebra.SesquilinearForm.Basic

Cited by

13 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidModuleCommSemiringAddCommMonoidModuleCommSemiringAddCommMonoidModule