Probably the most important class of vector norms, p-norms are noted ||x||p, where x is a vector, and is equal to:
||x||p = (sum |xi|p)^(1/p)
where the sum is on every element of the vector. The most common p-norm is of course 2-norm, which corresonds to the euclidian length of the vector. Every p-norm of the form above satisfies the requirements to be vector norms, that is:

  • ||x|| geq 0, and ||x|| = 0 only if x = 0;
  • ||x+y|| leq ||x|| + ||y||;
  • ||ax|| = |a| ||x||;
  • where the second condition is the triangle inequality and "a" in the third condition is any scalar. "geq" and "leq" respectively mean "greater or equal than" and "lower or equal than".