Mathematics is a language. This sentiment, expressed by the physicist Willard Gibbs, seems rather apt, especially when it comes to learning mathematics. Gaining fluency in mathematics and mathematical economics has much in common with learning a foreign language. Worrying too much about vocabulary lists and good grammar is a good way to kill your interest in the subject. Taking the plunge, trying to speak the language despite the errors you make early on, is not only more effective but also more enjoyable. This book is aimed at the economist willing to accept such a strategy of total immersion. This does not mean that I avoid the careful statement of assumptions, the crafting of rigorous proofs, and the like. Learning to do those things is an important part of any course in mathematical economics. What makes the approach adopted here distinctive is less a matter of substance than of style. Emphasizing economic intuition, I concentrate first and foremost on helping you develop a good ear for what the language of mathematical general equilibrium theory is trying to say. Learning the vocabulary and the grammar is easy once you can follow the conversation. What this means for the moment is that you should not worry too much if I seem to move rather quickly. You can always come back later to focus on the details.

The main goal of this chapter is to learn how to translate the competitive model of a pure exchange economy into the formal language of sets, functions, vector spaces, and linear functionals. This sounds more intimidating than it is.