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 ⊂ R {\displaystyle U\subset \mathbb {R} } to R {\displaystyle \mathbb {R} } under the norm ‖ ⋅ ‖ U {\displaystyle \|\cdot \|_{U}} defined by: ‖ f ‖ U = sup x ∈ U | f ( x ) | {\displaystyle \|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 ∈ U | f ( x ) 1 1 + x 2 | {\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 ‖ f ‖ = sup x ∈ U | f ( x ) ( 1 + x 4 ) | {\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 1 1 + x m {\displaystyle {\tfrac {1}{1+x^{m}}}} , the weighted space is called polynomial-weighted.
References
Kudryavtsev, L.D. (2001) [1994], "Weighted space", in Michiel Hazewinkel (ed.), Encyclopedia of Mathematics, EMS Press