314 CHAPTER 12. INNER PRODUCT SPACES, LEAST SQUARES
Corollary 12.3.2 Let A ∈L (X ,Y ) where X and Y are two inner product spaces of finitedimension or else Hilbert spaces. Then there exists a unique A∗ ∈L (Y,X) , the boundedlinear transformations, such that
(Ax,y)Y = (x,A∗y)X (12.5)
for all x ∈ X and y ∈ Y. The following formula holds
(αA+βB)∗ = αA∗+βB∗
Also, (A∗)∗ = A.
Proof: Let fy ∈L (X ,F) be defined as
fy (x)≡ (Ax,y)Y .
This is linear and ∣∣ fy (x)∣∣= |(Ax,y)Y | ≤ |Ax| |y| ≤ (||A|| |y|) |x|
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∗.
Finally, (A∗x,y) = (y,A∗x) = (Ay,x) = (x,Ay) while (A∗x,y) =(x,(A∗)∗ y
)and so for
all x, (x,(A∗)∗ y−Ay
)= 0
and so (A∗)∗ = A. ■