Number Theory: An Introduction to Mathematics

74 I The Expanding Universe of Numbers

Since‖w‖^2 =|γ 1 |^2 +···+|γm|^2 , this yieldsBessel’s inequality:

|〈v,e 1 〉|^2 +···+|〈v,em〉|^2 ≤‖v‖^2 ,

with strict inequality ifv/∈U.Foranyu∈U,wealsohave〈v−w,w−u〉=0and
so, by Pythagoras again,

‖v−u‖^2 =‖v−w‖^2 +‖w−u‖^2.

This shows thatwis the unique nearest point ofUtov.
From any linearly independent set of vectorsv 1 ,...,vm we can inductively
construct an orthonormal sete 1 ,...,emsuch thate 1 ,...,ekspan the same vector sub-
space asv 1 ,...,vkfor 1≤k≤m. We begin by takinge 1 =v 1 /‖v 1 ‖. Now suppose
e 1 ,...,ekhave been determined. If

w=vk+ 1 −〈vk+ 1 ,e 1 〉e 1 −···−〈vk+ 1 ,ek〉ek,

then〈w,ej〉= 0 (j= 1 ,...,k). Moreoverw=O,sincewis a linear combination of
v 1 ,...,vk+ 1 in which the coefficient ofvk+ 1 is 1. By takingek+ 1 =w/‖w‖, we obtain
an orthonormal sete 1 ,...,ek+ 1 spanning the same linear subspace asv 1 ,...,vk+ 1.
This construction is known asSchmidt’s orthogonalization process, because of its use
by E. Schmidt (1907) in his treatment of linear integral equations. The (normalized)
Legendre polynomials are obtained by applying the process to the linearly independent
functions 1,t,t^2 ,...in the spaceC(I),whereI=[− 1 ,1].
It follows that any finite-dimensional inner product spaceVhas an orthonormal
basise 1 ,...,enand that

‖v‖^2 =

∑n

j= 1

|〈v,ej〉|^2 for everyv∈V.

In an infinite-dimensional inner product spaceVan orthonormal setEmay even
be uncountably infinite. However, for a givenv∈V, there are at most countably many
vectorse∈Efor which〈v,e〉 =0. For if{e 1 ,...,em}is any finite subset ofEthen,
by Bessel’s inequality,

∑m

j= 1

|〈v,ej〉|^2 ≤‖v‖^2

and so, for eachn ∈ N, there are at mostn^2 −1 vectorse ∈ E for which
|〈v,e〉|>‖v‖/n.
If the vector subspaceUof all finite linear combinations of elements ofEis dense
inVthen, by the best approximation property of finite orthonormal sets,Pa rs ev a l ’s
equalityholds:
∑

e∈E

|〈v,e〉|^2 =‖v‖^2 for everyv∈V.

Parseval’s equality holds for the inner product spaceC(I),whereI=[0,1], and
the orthonormal setE={e^2 πint:n∈Z}since, byWeierstrass’s approximation the-
orem(see the references in§6 of Chapter XI), everyf∈C(I)is the uniform limit of
a sequence oftrigonometric polynomials. The result in this case was formally derived
by Parseval (1805).