# Definition:Dirac Delta Distribution

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## Definition

Let $a \in \R^d$ be a real vector.

Let $\phi \in \map \DD {\R^d}$ be a test function.

Let $\delta_a \in \map {\DD'} {\R^d}$ be a distribution.

Suppose $\delta_a$ is such that:

- $\map {\delta_a} \phi = \map \phi a$

Then $\delta_a$ is known as the **Dirac delta distribution**.

## Sources

- 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*: Chapter $\S 6.1$: A glimpse of distribution theory. Test functions, distributions, and examples