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.
