Linear Algebra, Theory And Applications, 2012a

292 INNER PRODUCT SPACES
⎛
⎞
m∑
m∑
m∑
= (z, x i )(y, x i ) − ⎝ (y, x i ) x i , (z,x j ) x j
⎠
i=1
i=1
i=1
j=1
m∑
m∑
= (z,x i )(y, x i ) − (y, x i ) (z, x i )=0.
This shows w given in (12.4) does minimize the function, z →|y − z| 2 for z ∈ M. It only
remains to verify uniqueness. Suppose than that w i ,i =1, 2 minimizes this function of z
for z ∈ M. Then from what was shown above,
|y − w 1 | 2 = |y − w 2 + w 2 − w 1 | 2
i=1
= |y − w 2 | 2 +2Re(y − w 2 ,w 2 − w 1 )+|w 2 − w 1 | 2
= |y − w 2 | 2 + |w 2 − w 1 | 2 ≤|y − w 2 | 2 ,
the last equal sign holding because w 2 is a minimizer and the last inequality holding because
w 1 minimizes.
12.3 Riesz Representation Theorem
The next theorem is one of the most important results in the theory of inner product spaces.
It is called the Riesz representation theorem.
Theorem 12.3.1 Let f ∈L(X, F) where X is an inner product space of dimension n.
Then there exists a unique z ∈ X such that for all x ∈ X,
f (x) =(x, z) .
Proof: First I will verify uniqueness. Suppose z j works for j =1, 2. Then for all x ∈ X,
0=f (x) − f (x) =(x, z 1 − z 2 )
and so z 1 = z 2 .
It remains to verify existence.
By Lemma 12.2.1, there exists an orthonormal basis,
{u j } n j=1 . Define z ≡
Then using Lemma 12.2.3,
(x, z) =
j=1
n∑
f (u j )u j .
j=1
⎛
⎞
n∑
⎝x, f (u j )u j
⎠ =
j=1
n∑
f (u j )(x, u j )
j=1
⎛
⎞
n∑
= f ⎝ (x, u j ) u j
⎠ = f (x) .
Corollary 12.3.2 Let A ∈L(X, Y ) where X and Y are two inner product spaces of finite
dimension. Then there exists a unique A ∗ ∈L(Y,X) such that
for all x ∈ X and y ∈ Y. The following formula holds
(Ax, y) Y
=(x, A ∗ y) X
(12.6)
(αA + βB) ∗ = αA ∗ + βB ∗