Banach’s Fixed Point Theorem

Keeping with this week’s theme of fixed point theory, today I’ll prove Banach’s Fixed Point Theorem. Banach’s Fixed Point Theorem asserts that any contractive mapping on a complete metric space admits a unique fixed point. To put it differently: take a map of the city you live in, crumple it up, and stomp it into the ground. Banach’s Fixed Point Theorem guarantees that there is a point on the map that is directly and exactly above the point it is representing.

Kakutani’s Fixed Point Theorem

Kakutani’s Fixed Point Theorem is a powerful generalization of Brouwer’s Fixed Point Theorem. It has several deep and important corollaries in economics, which include:

1. the Arrow-Debreu theorem, which proves the existence of a general equilibrium of an economy under certain assumptions. That is, there exists a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy. Both Arrow and Debreu went on to win Nobel Prizes in Economics in 1972 and 1983, respectively, for their work.
2. the existence of Nash equilibria of mixed games – which is arguably the most significant result in game theory. This earned John Nash the Nobel prize in 1994. In fact, Kakutani’s theorem reduced the original proof of Nash’s existence theorem from two-pages to a paragraph.

A second proof of Brouwer’s Theorem.

In my last post I mentioned that there are several proofs of the Brouwer Fixed Point Theorem, all of which are elegant. In this post, I’ll present a proof using degree theory. From what I understand, this was Brouwer’s original proof. After looking around a bit more, it seems that Brouwer’s understanding of a function’s degree was much more geometric and that the modern treatment of degree theory is either due to, or at least was best exposited by, John Milnor.

Brouwer’s Fixed Point Theorem

The Brouwer’s Fixed Point Theorem is a fundamental theorem in topology. In fact, when I took algebraic topology, it seemed like we proved it differently every few weeks. Indeed, the proofs using homotopy, homology, and degree theory are all elegant, but I will not present them in this post. Instead I will present a proof due to Emanuel Sperner that I find brilliant. The proof is based on Sperner’s Lemma, which I proved last time.

Sperner’s Lemma

Sperner’s Lemma is an elegant result in discrete mathematics. While it originated in the context of combinatorics and graph colorings, it also has surprisingly powerful applications to analysis and topology. In particular, it provides a combinatorial proof of the Intermediate Value Theorem (below) and the Brouwer Fixed Point Theorem (next time). Its powerful continuous corollaries have earned it the nickname:

“The Discrete Mathematician’s Intermediate Value Theorem.”