Theorems · Theorem · real analysis

sq_sum_le_card_mul_sum_sq

∀ {ι : Type u_1} {α : Type u_2} [inst : Semiring α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α] [ExistsAddOfLE α]
  {s : Finset ι} {f : ι → α}, (∑ i ∈ s, f i) ^ 2 ≤ ↑s.card * ∑ i ∈ s, f i ^ 2

Special case of Chebyshev's Sum Inequality or the Cauchy-Schwarz Inequality: The square of the sum is less than the size of the set times the sum of the squares.

Defined in

Mathlib.Algebra.Order.Chebyshev

Cited by

1 results in Mathlib

Foundations

Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringLinearOrderIsStrictOrderedRingExistsAddOfLE