distribution in analysis

广义函数

Analysis, Measure & PDE · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A distribution on an open set Omega in R^n is a continuous linear functional on C_c^infinity(Omega); locally, its action is bounded by finitely many derivative suprema of tests with support in a fixed compact set.

分析中的广义函数，是开集上紧支撑光滑测试函数空间的连续线性泛函；固定紧支集后，其作用可由有限多个导数的上确界控制。

Example

$\delta_2(\varphi)=\varphi(2)$ is a distribution, with $|\delta_2(\varphi)|\le\sup|\varphi|$.

Usage and distinctions

This sense of distribution is a generalized function, not a probability law; point evaluation need not come from an integrable function.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Mikko Salo — Fourier Analysis and Distribution Theory§3.7, definition of D′ and equation (3.18) · Checked 2026-10-14

Translation sources

1. 北京国际数学研究中心 — Enhanced Program for Graduate Study（2024）五、课程简介，4.微分方程基础选讲，（2）课程内容：广义函数和基本解 · Checked 2026-10-14

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