Weighted space

From Wikipedia, the free encyclopedia

In functional analysis, a weighted space is a space of functions under a weighted norm, which is a finite norm (or semi-norm) that involves multiplication by a particular function referred to as the weight.

Weights can be used to expand or reduce a space of considered functions. For example, in the space of functions from a set $U\subset\mathbb{R}$ to $\mathbb{R}$ under the norm $\|\cdot\|_U$ defined by

$\|f\|_U=\sup_{x\in U}{|f(x)|}$ ,

functions that have infinity as a limit point are excluded. However, the weighted norm $\|f\|=\sup_{x\in U}{|f(x)\frac{1}{1+x^2}|}$ is finite for many more functions, so the associated space contains more functions. Alternatively, the weighted norm $\|f\|=\sup_{x\in U}{|f(x)x^4|}$ is finite for many fewer functions.