146 CHAPTER 5. FUNCTIONS ON NORMED LINEAR SPACES

Lemma 5.12.2 If F is a set of functions which are upper semicontinuous, then g(x)≡inf{ f (x) : f ∈F} is also upper semicontinuous. Similarly, if F is a set of functions whichare lower semicontinuous, then if g(x) ≡ sup{ f (x) : f ∈F} it follows that g is lowersemicontinuous.

Note that in a metric space, the above definitions of upper and lower semicontinuity interms of sequences are equivalent to the definitions that

f (x)≥ limr→0

sup{ f (y) : y ∈ B(x,r)} , f (x)≤ limr→0

inf{ f (y) : y ∈ B(x,r)}

respectively.Here is a technical lemma which will make the proof of the saddle point theorem

shorter. It seems fairly interesting also.

Lemma 5.12.3 Suppose H : A×B→R is strictly convex in the first argument and con-cave in the second argument where A,B are compact convex nonempty subsets of Banachspaces E,F respectively and x→ H (x,y) is lower semicontinuous while y→ H (x,y) isupper semicontinuous. Let

H (g(y) ,y)≡minx∈A

H (x,y)

Then g(y) is uniquely defined and also for t ∈ [0,1] ,

limt→0

g(y+ t (z− y)) = g(y) .

Proof: First suppose both z,w yield the definition of g(y) . Then

H(

z+w2

,y)<

12

H (z,y)+12

H (w,y)

which contradicts the definition of g(y). As to the existence of g(y) this is nothing morethan the theorem that a lower semicontinuous function defined on a compact set achievesits minimum.

Now consider the last claim about “hemicontinuity”, continuity along a line. For allx ∈ A, it follows from the definition of g that

H (g(y+ t (z− y)) ,y+ t (z− y))≤ H (x,y+ t (z− y))

By concavity of H in the second argument,

(1− t)H (g(y+ t (z− y)) ,y)+ tH (g(y+ t (z− y)) ,z) (5.7)≤ H (x,y+ t (z− y)) (5.8)

Now let tn → 0. Does g(y+ tn (z− y))→ g(y)? Suppose not. By compactness, each ofg(y+ tn (z− y)) is in a compact set and so there is a further subsequence, still denoted bytn such that

g(y+ tn (z− y))→ x̂ ∈ A

Then passing to a limit in 5.8, one obtains, using the upper semicontinuity in one and lowersemicontinuity in the other the following inequality.

H (x̂,y)≤ lim infn→∞

(1− tn)H (g(y+ tn (z− y)) ,y)+