Metric Spaces

The topic of today's blog is the concept of a Metric Space. A metric space is a set of elements(in most cases numbers) that are endowed with a "metric" that satisfies certain conditions. The "metric" is simply a function of two variables on the set of elements. It will be denoted d(x,y). For metric spaces, the metric d must satisfy the following properties:

1. d(x,y)>0 for all inequal x,y
2. d(x,y)=0 if and only if x=y
3. d(x,y)=d(y,x)
4. d(x,z)< or =d(x,y)+d(y,z)

In words, the conditions state that the distance function is positively defined(it is never negative)and symmetric (d(x,y)=d(y,x)). Also, d satisfies the triangle inequality. One elementary example of a metric space is the set of real numbers R endowed with the absolute value function d(x,y)=absolute value of (x-y). In all high school and lower math classes, and some college classes it is subconsciously assumed that when dealing with real numbers, the absolute value function is the metric that is given for R.However, this function is not the only possible metric on R, take for example the function:

d(x,y)=0 when x=y,

d(x,y)=1 when x is inequal to y

This function satisfies properties 1-4, because it is never negative, only 0 when x=y, symmetric, and not in violation of the triangle inequality. This function is called the discrete metric, and is a metric for any set of elements(the absolute value function, on the other hand, is not, because there is no definition for the absolute value of the difference of two functions). In addition the absolute value metric and the discrete metric, there are an infinite number of metrics on R, as well as any set of elements.

Why study metric spaces? Metric spaces are essential to the further study of calculus. One big part of calculus deals with entire functions as points in a space, instead of singular numbers as points, and the functions on these so called "function spaces". Taking limits, derivatives, and integrals rigorously all require the use of metric spaces. The most general definitions of limits, continuity, open and closed sets, and compactness, completeness, and convergence are all formulated in the language of metric spaces. The terms listed are all central to calculus, and in fact, calculus can be thought of as based on four of the terms: continuity, compactness, completeness, and convergence. Without metric spaces, calculus would be nowhere as sophisticated as it is today.

Brouwer Fixed Point Theorem

This blog post will be on a very interesting theorem called the Brouwer Fixed Point Theorem, focusing on the case for the 2 dimensional disk, otherwise known as the circle (with all points inside it included). The Brouwer Fixed Point Theorem for the 2 dimensional disk states that every continuous mapping f from the disk to itself has a fixed point, e.g. a point x such that f(x)=x. One can visualize this theorem by stirring a cup of coffee; the point that does not move when the cup is stirred is the center point. Similarly, the eye of a hurricane is the one place where no winds are swirling.

First of all, the definition of a function, continuous function and a function from the disk to itself should be precisely defined. A function is a relation that given an element, assigns one and only one element. Elements are commonly thought of as points, but they could also be entire functions, or other, more general concepts. A continuous function is a function that over its entire domain (all possible x values), has the following property: as x approaches a, f(x) approaches f(a). Intuitively speaking, the function is "smooth" and has no "jumps" in it-values of x that are very close together are assigned values of f(x) that are also very close together.Graphs of continuous functions can be drawn without ever lifting the writing instrument. There are other, even more precise definitions of continuity, but this one is sufficient. Finally, a function that maps the disk to itself is simply a function whose domain is the disk and whose range (all assigned values) does not include any points outside of the disk.

The proof of the theorem is fairly complicated, and will be omitted. Rather, applications of the theorem will be discussed. John Nash used a version of this fixed point theorem to prove the existence of the Nash Equilibrium, which is when in a game of two or more parties, each party individually makes the best decision it can make taking into account the decisions that the other parties can make. This finding won Nash the Nobel Prize in Economics. The Fixed Point Theorem is also used to prove the existence of other equilibria in economics, such as market equilibrium.

The Brouwer Fixed Point Theorem applies also for any line segment, and can be generalized to the n dimensional sphere. The theorem also applies to any surface, or in higher dimensions, "manifold" that is similar enough to the sphere. The meaning of "similar enough" is rigorously established in a field of mathematics called "Topology," which deals with the distinct shapes and figures of various manifolds.