Mathematics Prelims

July 14, 2008

Introduction to Metric Spaces

Filed under: Analysis,Functional Analysis — cjohnson @ 1:16 pm

A metric space is a pair $(X, d)$ where $X$ is a set and d is a function $d : X \times X \to \mathbb{R}$ where the following properties hold for all $x, y, z \in X$.

• $0 \leq d(x, y) < \infty$
• $d(x, y) = 0 \Leftrightarrow x = y$
• $d(x, y) = d(y, x)$
• $d(x, z) \leq d(x, y) + d(y, z)$

The value d(x, y) is referred to as the distance between x and y.  The four properties of a metric tell us that the distance between any two points can’t be negative or infinite; the distance is zero if and only if the two points are the same; distances are symmetric; the distance between any two points is less than or equal to the distance from the first point to some other point, then to our second point.  This last property is called the triangle inequality and in practice is more difficult to show for a space than the other three properties.

Common examples of metric spaces are the real line where $d(x, y) = |x - y|$, the real plane $\mathbb{R}^2$ where $d(x, y) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2}$ (where x = (x_1, x_2) and similarly for y).  Other examples of metric sapces include C[a, b], the set of all continuous functions on the closed interval from a to b, where the metric is given by $d(f, g) = \max_{t \in [a, b]} |f(t) - g(t)|$, and the set of all bounded functions, denoted $l^\infty$, where $d(x, y) = \sup_{n \in \mathbb{N}} |x_n - y_n|$.

A natural question to ask is whether or not a metric can be defined on any arbitrary set.  We can, in fact, always use the so-called discrete metric on any set.  The discrete metric is defined to be $d(x, y) = 0$ whenever $x = y$, and $d(x, y) = 1$ if $x \neq y$.

An important example of metric spaces are the $l^p$ spaces.  In coming posts we’ll introduce three theorems then use these theorems to show that the $d(x, y) = ( \sum_{n=1}^\infty |x_n - y_n|^p )^{\frac{1}{p}}$ is a metric on the l^p spaces.