5.12. SADDLE POINTS∗ 149

Thus, letting C ≡max{∥x∥ : x ∈ A}

εC2 +maxy∈B

minx∈A

H (x,y)≥minx∈A

maxy∈B

H (x,y)

Since ε is arbitrary, it follows that

maxy∈B

minx∈A

H (x,y)≥minx∈A

maxy∈B

H (x,y)

This proves the first part because it was shown above in 5.10 that

minx∈A

maxy∈B

H (x,y)≥maxy∈B

minx∈A

H (x,y)

Now consider 5.9 about the existence of a “saddle point” given the equality of minmaxand maxmin. Let

α = maxy∈B

minx∈A

H (x,y) = minx∈A

maxy∈B

H (x,y)

Then fromy→min

x∈AH (x,y) and x→max

y∈BH (x,y)

being upper semicontinuous and lower semicontinuous respectively, there exist y0 and x0such that

α = minx∈A

H (x,y0) =

.

maxy∈B

minimum of u.s.cminx∈A

H (x,y) = minx∈A

maximum of l.s.c.maxy∈B

H (x,y) = maxy∈B

H (x0,y)

Thenα = max

y∈BH (x0,y)≥ H (x0,y0) , α = min

x∈AH (x,y0)≤ H (x0,y0)

so in fact α = H (x0,y0) and from the above equalities,

H (x0,y0) = α = minx∈A

H (x,y0)≤ H (x,y0)

H (x0,y0) = α = maxy∈B

H (x0,y)≥ H (x0,y)

and so H (x0,y) ≤ H (x0,y0) ≤ H (x,y0) . Thus if the minmax condition holds, then thereexists a saddle point, namely (x0,y0).

Finally suppose there is a saddle point (x0,y0) where

H (x0,y)≤ H (x0,y0)≤ H (x,y0)

Then

minx∈A

maxy∈B

H (x,y)≤maxy∈B

H (x0,y)≤ H (x0,y0)≤minx∈A

H (x,y0)≤maxy∈B

minx∈A

H (x,y)

However, as noted above, it is always the case that

maxy∈B

minx∈A

H (x,y)≤minx∈A

maxy∈B

H (x,y) ■

What was really needed? You needed compactness of A,B and these sets needed to be ina linear space. Of course there needed to be a norm for which x→∥x∥ is strictly convex andlower semicontinuous, so the conditions given above are sufficient but maybe not necessary.You might try generalizing this much later after reading about weak topologies.