Suppose $f, g\in S=L^p([0,1],\Sigma,\mu,[0,1])$. The objective $L:S\times S\to R$ is given by
$$L(f,g)= \int f (h-g) d\mu, $$
where $h\in S$ is fixed. Could we apply Sion Minmax theorem to conclude that $$ \inf_{f\in S} \sup_{g\in S} L(f,g)=\sup_{g\in S} \inf_{f\in S} L(f,g) $$
because $L(g,⋅)$ is upper semicontinuous and quasi-concave on $S$ for each $g∈S$ and $L(⋅,f)$ is lower semicontinuous and quasi-convex on S for each $f\in S$. Is it right to say $L$ is linear in $g$ and $f$?
I think if here $L^p=L^2$, then L is continuous in each argument and thus both upper and lower semicontinuous because it is a duality product?
Thanks.