Advanced book on Mathematics Olympiad

2.1 Identities and Inequalities 33

(i)〈x, x〉≥0, with equality if and only ifx=0,
(ii)〈x, y〉=〈y, x〉, for any vectorsx, y(here the bar stands for complex conjugation if
the vector space is complex),
(iii)〈λ 1 x 1 +λ 2 x 2 ,y〉=λ 1 〈x 1 ,y〉+λ 2 〈x 2 ,y〉, for any vectorsx 1 ,x 2 ,yand scalarsλ 1
andλ 2.
The quantity‖x‖=

√

〈x, x〉is called the norm ofx. Examples of inner product
spaces areRnwith the usual dot product,Cnwith the inner product

〈(z 1 ,z 2 ,...,zn), (w 1 ,w 2 ,...,wn)〉=z 1 w 1 +z 2 w 2 +···+znwn,

but also the space of square integrable functions on an interval[a, b]with the inner
product

〈f, g〉=

∫b

a

f(t)g(t)dt.

The Cauchy–Schwarz inequality.Letx, ybe two vectors. Then

‖x‖·‖y‖≥|〈x, y〉|,

with equality if and only if the vectorsxandyare parallel and point in the same direction.

Proof.We have

0 ≤〈‖y‖x−‖x‖y,‖y‖x−‖x‖y〉= 2 ‖x‖^2 ‖y‖^2 −‖x‖‖y‖(〈x, y〉+〈y, x〉),

hence 2‖x‖·‖y‖≥(〈x, y〉+〈y, x〉). Yet another trick: rotateyby〈x, y〉/|〈x, y〉|. The
left-hand side does not change, but because of property (ii) the right-hand side becomes
1
|〈x,y〉|(〈x, y〉〈x, y〉+〈x, y〉〈x, y〉), which is the same as 2|〈x, y〉|. It follows that

‖x‖·‖y‖≥|〈x, y〉|,

which is the Cauchy–Schwarz inequality in its full generality. In our sequence of deduc-
tions, the only inequality that showed up holds with equality precisely when the vectors
are parallel and point in the same direction. 

As an example, iff andgare two complex-valued continuous functions on the
interval[a, b], or more generally two square integrable functions, then

∫b

a

|f(t)|^2 dt

∫b

a

|g(t)|^2 dt≥

∣

∣∣

∣

∫b

a

f(t)g(t)dt

∣

∣∣

∣

2

.

Let us turn to more elementary problems.