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 to R under the norm ∥ ⋅ ∥ U defined by: ∥ f ∥ U = sup x ∈ 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 | is finite for many more functions, so the associated space contains more functions. Alternatively, the weighted norm ∥ f ∥ = sup x ∈ U | f ( x ) x 4 | is finite for many fewer functions.
When the weight is of the form 1 1 + x m , the weighted space is called polynomial-weighted.
