We prove the almost equivalence between two-player zero-sum games and conic linear programming problems in reflexive Banach spaces. The previous fundamental results of von Neumann, Dantzig, Adler, and von Stengel, regarding the equivalence between finite games with strategy sets defined over $\mathbb{R}^n$, and linear programming, are therefore generalized to the infinite-dimensional case. In fact, we show that for every two-player zero-sum game with a bilinear function of the form $u(x,y)=\langle y,Ax\rangle$, for some linear operator $A$, and strategy sets that represent bases of convex cones, the minimax theorem holds, and its game value and Nash equilibria can be computed by solving a primal-dual pair of conic linear problems. Conversely, the minimax theorem for the same class of games "almost always" implies strong duality of conic linear programming. The main results are applied to a number of infinite zero-sum games, whose classes include those of semi-infinite, semidefinite, time-continuous, quantum, polynomial, and homogeneous separable games.

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