134 CHAPTER 5. FUNCTIONS ON NORMED LINEAR SPACES
Definition 5.11.4 A norm is called strictly convex if whenever x ̸= y,∥∥∥∥x+ y2
∥∥∥∥< ∥x∥2 +∥y∥2
Theorem 5.11.5 Let E,F be Banach spaces with E having a strictly convex norm.Also suppose that A ⊆ E,B ⊆ F are compact and convex sets and that H : A×B→ R issuch that
x→ H (x,y) is convex
y→ H (x,y) is concave
Assume that x→ H (x,y) is lower semicontinuous and y→ H (x,y) is upper semicontin-uous. Then minx∈A maxy∈B H (x,y) = maxy∈B minx∈A H (x,y) . This condition is equivalentto the existence of (x0,y0) ∈ A×B such that
H (x0,y)≤ H (x0,y0)≤ H (x,y0) for all x,y (5.8)
called a saddle point.
Proof: One part of the main equality is obvious.
maxy∈B
H (x,y)≥ H (x,y)≥minx∈A
H (x,y)
and so for each x, maxy∈B H (x,y)≥maxy∈B minx∈A H (x,y) and so
minx∈A
maxy∈B
H (x,y)≥maxy∈B
minx∈A
H (x,y) (5.9)
Next consider the other direction.Define Hε (x,y)≡ H (x,y)+ ε ∥x∥2 where ε > 0. Then Hε is strictly convex in the first
variable. This results from the observation that∥∥∥∥x+ y2
∥∥∥∥2
<
(∥x∥+∥y∥
2
)2
≤ 12
(∥x∥2 +∥y∥2
),
By Lemma 5.11.3 there exists a unique x ≡ g(y) with Hε (g(y) ,y) ≡ minx∈A Hε (x,y) andalso, whenever y,z ∈ A, limt→0+ g(y+ t (z− y)) = g(y). Thus
Hε (g(y) ,y) = minx∈A
Hε (x,y) .
But also this shows that y→ Hε (g(y) ,y) is the minimum of functions which are uppersemicontinuous and so this function is also upper semicontinuous. Hence there exists y∗
such thatmaxy∈B
Hε (g(y) ,y) = Hε (g(y∗) ,y∗) = maxy∈B
minx∈A
Hε (x,y) (5.10)
Thus from concavity in the second argument and what was just defined, for t ∈ (0,1) ,
Hε (g(y∗) ,y∗)≥ Hε (g((1− t)y∗+ ty) ,(1− t)y∗+ ty)
≥ (1− t)Hε (g((1− t)y∗+ ty) ,y∗)+ tHε (g((1− t)y∗+ ty) ,y)
≥ (1− t)Hε (g(y∗) ,y∗)+ tHε (g((1− t)y∗+ ty) ,y) (5.11)