Theorems · Theorem · linear algebra

LinearMap.BilinForm.apply_mul_apply_le_of_forall_zero_le

∀ {R : Type u_1} {M : Type u_5} [inst : CommRing R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R]
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] (B : LinearMap.BilinForm R M),
  (∀ (x : M), 0 ≤ (B x) x) → ∀ (x y : M), (B x) y * (B y) x ≤ (B x) x * (B y) y

The Cauchy-Schwarz inequality for positive semidefinite forms.

Defined in

Mathlib.LinearAlgebra.SesquilinearForm.Basic

Cited by

2 results in Mathlib

Foundations

Depth 45 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLinearOrderIsStrictOrderedRingAddCommGroupModule