On Proposition 2.5 in “An Introduction to the Theory of Mechanism Design”

An elegant proof using normed spaces was given to Proposition 2.5 in “An Introduction to the Theory of Mechanism Design”concering the existence of the direct mechanism which maximizes the seller's expected revenue among all incentive-compatible and individually rational direct mechanisms. Since this proposition is the first main result and normed spaces are not so easy mathematical tool compared with those used later, we felt the need for an easier proof. The outline of the proof of our result in this note is almost same as that of Proposition 3.4 in the book. We also give a condition under which we have the uniqueness of the real number that characterizes the objective direct mechanism.