276 CHAPTER 11. INNER PRODUCT SPACES

The vectors, {uj}nj=1 , generated in this way are therefore an orthonormal basis becauseeach vector has unit length. ■

The process by which these vectors were generated is called the Gram Schmidt process.The following corollary is obtained from the above process.

Corollary 11.2.2 Let X be a finite dimensional inner product space of dimension n whosebasis is {u1, · · · , uk, xk+1, · · · , xn} . Then if {u1, · · · , uk} is orthonormal, then the GramSchmidt process applied to the given list of vectors in order leaves {u1, · · · , uk} unchanged.

Lemma 11.2.3 Suppose {uj}nj=1 is an orthonormal basis for an inner product space X.Then for all x ∈ X,

x =

n∑j=1

(x, uj)uj .

Proof: Since {uj}nj=1 is a basis, there exist unique scalars {αi} such that

x =

n∑j=1

αjuj

It only remains to identify αk. From the properties of the inner product,

(x, uk) =

n∑j=1

αj (uj , uk) =

n∑j=1

αjδjk = αk ■

The following theorem is of fundamental importance. First note that a subspace of aninner product space is also an inner product space because you can use the same innerproduct.

Theorem 11.2.4 Let M be a finite dimensional subspace of X, an inner product space andlet {ei}mi=1 be an orthonormal basis for M . Then if y ∈ X and w ∈M,

|y − w|2 = inf{|y − z|2 : z ∈M

}(11.2)

if and only if(y − w, z) = 0 (11.3)

for all z ∈M. Furthermore,

w =

m∑i=1

(y, xi)xi (11.4)

is the unique element of M which has this property. It is called the orthogonal projection.

Proof: First we show that if 11.3, then 11.2. Let z ∈M be arbitrary. Then

|y − z|2 = |y − w + (w − z)|2

= (y − w + (w − z) , y − w + (w − z))

= |y − w|2 + |z − w|2 + 2Re (y − w,w − z)

The last term is given to be 0 and so

|y − z|2 = |y − w|2 + |z − w|2