SupNorm

Supnorm, also known as the supremum norm or infinity norm, is a concept used in functional analysis and numerical analysis. It is a type of norm defined for functions or vectors. For a continuous function f defined on a closed and bounded set, the supnorm is the maximum absolute value of the function over that set. Mathematically, for a function $f$ on a domain $D$, the supnorm is denoted as $||f||_{\infty} = \sup_{x \in D} |f(x)|$.

This norm is particularly useful because it measures the largest deviation of a function from zero. In the context of vectors, the supnorm is equivalent to the maximum absolute value of any component of the vector. For a vector $v = (v_1, v_2, ..., v_n)$, the supnorm is $||v||_{\infty} = \max(|v_1|, |v_2|, ..., |v_n|)$.

The supnorm possesses the properties of a mathematical norm: it is non-negative, zero only for the zero function or zero vector, homogeneous (scaling a function or vector scales its supnorm by the same factor), and satisfies the triangle inequality (the supnorm of the sum of two functions or vectors is less than or equal to the sum of their individual supnorms). Its simplicity and effectiveness in bounding function behavior make it a fundamental tool in various areas of mathematics and its applications.