| # Definition: vector norm Informally, a (vector) norm is a function which assigns a length to vectors. Any sensible measure of length should satisfy the following basic properties: it should be a convex function of its argument (that is, the length of an average of two vectors should be always less than the average of their lengths); it should be positive-definite (always non-negative, and zero only when the argument is the zero vector), and preserve positive scaling (so that multiplying a vector by a positive number scales its norm accordingly). Formally, a vector norm is a function which satisfies the following properties. Definition of a vector norm Positive homogeneity: for every , , we have . Triangle inequality: for every , we have Definiteness: for every , implies . A consequence of the first two conditions is that a norm only assumes non-negative values, and that it is convex. Popular norms include the so-called -norms, where or : with the convention that when , .