This chapter characterizes the main theoretical difficulties in the proof of the existence of competitive equilibrium in infinite dimensional models. The existence of Walrasian equilibrium can be established in considerable generality. The well known difficulties that arise in finite dimensional cases to prove the existence of equilibrium, like the non-convexity of preferences or other of this kind, survive in infinite dimensional cases, and at the same time new problems appear. The economic problem determines the mathematical tool that is applied to obtain the solution and its precise formulation; this does not mean that lecturers have to take a mathematical tool and then look for application. While in finite dimensional vector spaces there is only one Hausdorff linear topology, an arbitrary infinite dimensional vector space admits more than one linear topology. Each space of a dual pair can be interpreted as a set of linear functionals on the other.