# How do I interpret the definition of a distribution in this book by Joshi & Friedlander?

I was wondering about the following definition in the book Introduction to the Theory of Distributions, by Joshi and Friedlander:

In the Appendix the bracket operation is defined as

To me this definition of a distribution doesn't make sense if we view the bracket operation as an inner product, e.g., an integral, since the domain of $$u$$ is $$C_c^\infty(X)$$ whereas the domain of $$\phi$$ is $$X$$. So then how do we make sense of $$\int u(x)\phi(x) dx$$?

I would understand if $$u$$ were, say, a continuous function from $$X$$ to $$\mathbb{R}$$. Then $$u$$ would determine a linear form through $$\langle u, \phi \rangle$$. Is that what the definition means? We take some object $$u$$, then $$u$$ determines a linear form through the pairing $$\langle u, \phi \rangle$$? And then we refer to $$u$$ as a linear form because somehow the linear form determines $$u$$ uniquely? Perhaps this bit from the book is relevant:

• $\langle u, \phi \rangle$ is just a notation for the value of $u$ at $\phi$. No inner product is involved here. Commented Feb 9, 2021 at 6:14
• You seem to be assuming that every linear form on $C_c^\infty(\mathbb{R}^n)$ comes from a function. But there are other forms, and those are called distributions. Commented Feb 9, 2021 at 6:43

## 1 Answer

The notation $$\langle u,\varphi\rangle$$ is just a fancy way of writing $$u(\varphi)$$ for a linear from $$u\colon C^\infty_c(X)\to\mathbb{C}$$. Examples of linear forms which are not defined by functions are $$\delta_0$$ or $$\partial^\alpha\delta_0$$ which are defined as $$\langle \delta_0, \varphi\rangle = \varphi(0) \qquad \text{and}\qquad \langle \partial^\alpha\delta_0, \varphi\rangle = (-1)^{|\alpha|}(\partial^\alpha\varphi)(0).$$ For these two it is easy to check that they satisfy the condition stated above. A slightly more complicated example might be $$\langle \mathrm{vp}\frac{1}{x}, \varphi\rangle = \lim_{\varepsilon\to 0} \int_{\mathbb{R}\setminus(-\varepsilon,\varepsilon)} \frac{\varphi(x)}{x}\,\mathrm{d}x.$$ Note that none of these distributions can be represented by a continuous function.