supremumnormi

The supremum norm, also known as the L-infinity norm or the uniform norm, is a specific way to measure the "size" or "magnitude" of a function or a vector. For a function f defined on a set X, the supremum norm is the smallest upper bound of the absolute values of f over X. In simpler terms, it's the maximum absolute value that the function takes anywhere within its domain. Mathematically, it is often denoted as ||f||_infinity or ||f||_sup.

For a vector x in a finite-dimensional space, like R^n, the supremum norm is simply the maximum absolute value of its components. For example, if x = (x_1, x_2, ..., x_n), then ||x||_infinity = max(|x_1|, |x_2|, ..., |x_n|). This norm is particularly useful in analysis, functional analysis, and numerical methods because it captures the worst-case scenario or the peak value of a function or vector. It's a fundamental concept in understanding convergence of sequences of functions, as convergence in the supremum norm implies uniform convergence.