Weighted space

In mathematics, specifically functional analysis, a weighted space is a function space equipped with a weighted norm, which is a finite norm (or semi-norm) that involves multiplication by a particular weight function.

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 $\Vert \cdot {\Vert }_U$ defined by $\Vert f{\Vert }_U=\underset{x\in U}{\sup }|f(x)|$, functions that have infinity as a limit point are excluded. However, the weighted norm

$\Vert f\Vert =\underset{x\in U}{\sup }\left|\frac{f(x)}{1+x^2}\right|$

is finite for many more functions, so the associated space contains more functions. Alternatively, the weighted norm

$\Vert f\Vert =\underset{x\in U}{\sup }|f(x)(1+x^4)|$

is finite for many fewer functions. When the weight is of the form $\frac{1}{1+x^m}$, the weighted space is called polynomial-weighted.

References

• Hazewinkel, Michiel, ed. (2001), "Weighted space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page