Page 54 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Convex analysis 41

∗
(in general, f takes the value +∞ even if f takes only finite values) where h.; .i
n
denotes the scalar product in R .
n
(ii) The bidual of f is the function f ∗∗ : R → R ∪ {±∞} defined by

∗
f ∗∗ (x)= sup {hx; x i − f (x )} .
∗
∗
x ∗ ∈R n
Let us see some simple examples that will be studied in Exercise 1.5.4.
p
Example 1.53 (i) Let n =1 and f (x)= |x| /p,where 1 <p < ∞.We then
find
1 ∗ p 0
∗
∗
f (x )= |x |
p 0
where p is,as usual,defined by 1/p +1/p =1.
0
0
¡ ¢ 2
2
(ii) Let n =1 and f (x)= x − 1 .We then have
⎧ ¡ ¢ 2
2
⎨ x − 1 if |x| ≥ 1
f ∗∗ (x)=
⎩
0 if |x| < 1 .
(iii) Let n =1 and
⎧
0 if x ∈ (0, 1)
⎨
f (x)=
+∞ otherwise.
⎩
We immediately find that
⎧
⎨ x ∗ if x ≥ 0
∗
∗
f (x )= sup {xx } =
∗
∗
x∈(0,1) ⎩ 0 if x ≤ 0
∗
∗
f is often called the indicator function of (0, 1),and f the support function.
We also have ⎧
0 if x ∈ [0, 1]
⎨
f ∗∗ (x)=
⎩
+∞ otherwise
and hence f ∗∗ is the indicator function of [0, 1].
(iv) Let X ∈ R 2×2 ,where R 2×2 is the set of 2 × 2 real matrices which will be
4
identified with R ,and let f (X)= det X,then
∗
∗
f (X ) ≡ +∞ and f ∗∗ (X) ≡−∞ .