Euclidean Vector Spaces

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. In this definition, no operation between points in a metric space is required. However, i our course on vector spaces, there is an internal operation (called addition) between its elements is imposed. In order to incorporate metric and addition between vectors, a special metric function, known as norm, is defined.

The most famous metric in history of mathematics is of course the Euclidean metric. In three-dimensional space, the Euclidean distance is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem. Once the Cartesian system of coordinates in a vector space is established, the Euclidean metric can be defined. Therefore, ℝⁿ or ℂⁿ automatically become Euclidean spaces with metrics generated by the dot products.