11.3. RIESZ REPRESENTATION THEOREM 279

Then using Lemma 11.2.3,

(x, z) =

x, n∑j=1

f (uj)uj

 =

n∑j=1

f (uj) (x, uj)

= f

 n∑j=1

(x, uj)uj

 = f (x) . ■

Corollary 11.3.2 Let A ∈ L (X,Y ) where X and Y are two inner product spaces of finitedimension. Then there exists a unique A∗ ∈ L (Y,X) such that

(Ax, y)Y = (x,A∗y)X (11.5)

for all x ∈ X and y ∈ Y. The following formula holds

(αA+ βB)∗= αA∗ + βB∗

Proof: Let fy ∈ L (X,F) be defined as

fy (x) ≡ (Ax, y)Y .

Then by the Riesz representation theorem, there exists a unique element of X, A∗ (y) suchthat

(Ax, y)Y = (x,A∗ (y))X .

It only remains to verify that A∗ is linear. Let a and b be scalars. Then for all x ∈ X,

(x,A∗ (ay1 + by2))X ≡ (Ax, (ay1 + by2))Y

≡ a (Ax, y1) + b (Ax, y2) ≡

a (x,A∗ (y1)) + b (x,A∗ (y2)) = (x, aA∗ (y1) + bA∗ (y2)) .

Since this holds for every x, it follows

A∗ (ay1 + by2) = aA∗ (y1) + bA∗ (y2)

which shows A∗ is linear as claimed.Consider the last assertion that ∗ is conjugate linear.(

x, (αA+ βB)∗y)≡ ((αA+ βB)x, y)

= α (Ax, y) + β (Bx, y) = α (x,A∗y) + β (x,B∗y)

= (x, αA∗y) +(x, βA∗y

)=(x,(αA∗ + βA∗) y) .

Since x is arbitrary,(αA+ βB)

∗y =

(αA∗ + βA∗) y

and since this is true for all y,

(αA+ βB)∗= αA∗ + βA∗. ■

Definition 11.3.3 The linear map, A∗ is called the adjoint of A. In the case when A : X →X and A = A∗, A is called a self adjoint map. Such a map is also called Hermitian.