Using a theorem of König [10] and a theorem proved earlier [7] minimax theorems for the discrete mixed extension of two-person zero-sum games are derived. It is shown that the problem of approximating a normed linear functional by an integral can be treated by game theoretic means. Finally, an example of such an approximation theorem is used for a simple proof of the Krein-Smulian and the Eberlein-Šmulian theorem.