Weighted space

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

单词	Weighted space
释义	Weighted space 英语百科 Weighted space 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 $U\subset\mathbb{R}$ to $\mathbb{R}$ under the norm $\\|\cdot\\|_U$ defined by: $\\|f\\|_U=\sup_{x\in U}{\|f(x)\|}$ , functions that have infinity as a limit point are excluded. However, the weighted norm $\\|f\\|=\sup_{x\in U}{\left\|f(x)\tfrac{1}{1+x^2}\right\|}$ is finite for many more functions, so the associated space contains more functions. Alternatively, the weighted norm $\\|f\\|=\sup_{x\in U}{\left\|f(x)x^4\right\|}$ is finite for many fewer functions.
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