Another famous inequality is given various names in different texts, although in the United States it is usually referred to as the Cauchy - Schwarz Inequality (named for Augustin Cauchy, 1789-1857; and Hermann Amandus Schwarz, 1843-1921). Some call it the Schwarz Inequality, while others, including the Russians, call it the Cauchy-Buniakowski Inequality.

THEOREM: Let a₁, a₂, ..., a_n and b₁, b₂, ..., b_n be any real numbers. Then

that is,

Proof: We define a polynomial function f(x) by
 

Clearly f(x) is positive or zero for all real numbers x, since it is a sum of squares. Now

Let  so that

f(x) = Ax² +2Bx + C. Since for all x, the discriminant since D > 0 implies that f(x) is sometimes positive and sometimes negative. (See Section 10.2.) Now
 

Hence  Translating this back into our original notation, we have the Cauchy-Schwarz Inequality:

Monday, June 22, 1998