# Continuous space and metrics

As most of this chapter's content will be dealing with trying to predict or optimize continuous variables, let's first understand how to measure the difference in a continuous space. Unless a drastically new discovery is made pretty soon, the space we live in is a three-dimensional Euclidian space. Whether we like it or not, this is the world we are mostly comfortable with today. We can completely specify our location with three continuous numbers. The difference in locations is usually measured by distance, or a metric, which is a function of a two arguments that returns a single positive real number. Naturally, the distance, , between *X* and *Y* should always be equal or smaller than the sum of distances between *X* and *Z* and *Y* and *Z*:

For any *X*, *Y*, and *Z*, which is also called triangle inequality. The two other properties of a metric is symmetry:

Non-negativity of distance:

Here, the metric is `0`

if, and only if, *X=Y*. The distance is the distance as we understand...