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 ${\displaystyle U\subset \mathbb {R} }$ to ${\displaystyle \mathbb {R} }$ under the norm ${\displaystyle \|\cdot \|_{U}}$ defined by: ${\displaystyle \|f\|_{U}=\sup _{x\in U}{|f(x)|}}$, functions that have infinity as a limit point are excluded. However, the weighted norm ${\displaystyle \|f\|=\sup _{x\in U}{\left|f(x){\tfrac {1}{1+x^{2}}}\right|}}$ is finite for many more functions, so the associated space contains more functions. Alternatively, the weighted norm ${\displaystyle \|f\|=\sup _{x\in U}{\left|f(x)(1+x^{4})\right|}}$ is finite for many fewer functions.

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

References

• Kudryavtsev, L.D. (2001) [1994], "Weighted space", in Michiel Hazewinkel (ed.), Encyclopedia of Mathematics, EMS Press