Weighted space

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 $$\|f\|=\sup_{x\in U}{\left|f(x)\tfrac{1}{1+x^2}\right|}$$ is finite for many more functions, so the associated space contains more functions. Alternatively, the weighted norm $$\|f\|=\sup_{x\in U}{\left|f(x)(1 + x^4)\right|}$$ is finite for many fewer functions.

When the weight is of the form $$\tfrac{1}{1+x^m}$$, the weighted space is called polynomial-weighted.