Chapters 6 through 9 present a survey of all of the mathematics used in Chapters 10 through 22 – the mathematics needed to describe an economy with continuous supply and demand functions. Many of the topics treated here are part of the usual content of an introductory course on analysis in RN: sets, limits, convergence, open and closed sets, and continuous functions. In addition, there are topics that often are not prominent in the course on real analysis that turn out to be central to mathematical economics: convexity, separation theorems, fixed-point theorems, the Shapley-Folkman Theorem. This part assumes the student is familiar with the notation and concepts of analytic geometry. It is not a substitute for a course in real analysis (to which the student is strongly recommended).
Prof. Debreu (1986) reminds us of the distinctive usefulness of Euclidean N-dimensional space:
[Economics's] central concepts, commodity and price, are quantified in a unique manner, as soon as units of measurement are chosen. Thus for an economy with a finite number of commodities, the action of an economic agent is described by listing his input, or his output, of each commodity. Once a sign convention distinguishing inputs from outputs is made, the action of an agent is represented by a point in the commodity space, a finite-dimensional real vector space. […]