L^p Space - The p-norm in Finite Dimensions

The p-norm in Finite Dimensions

The length of a vector x = (x₁, x₂, …, x_n) in the n-dimensional real vector space Rn is usually given by the Euclidean norm:

The Euclidean distance between two points x and y is the length of the straight line between the two points. In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space. For example, taxi drivers in Manhattan should measure distance not in terms of the length of the straight line to their destination, but in terms of the Manhattan distance, which takes into account that streets are either orthogonal or parallel to each other. The class of p-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science.