Viewing distributions as "generalized functions"

Question

A distribution is defined to be element of the dual space of $C_0^\infty$. Since a distribution is thus a linear functional, I would think that a distribution is itself a function but this is not the case as can be seen by the Dirac-$\delta$ function. I am thus a little unclear on what a distribution is exactly if it's somehow more than a linear functional. What do we exactly mean by saying they are "generalized functions", why not just view them as linear functionals?

Answer 1

We do view distributions as mappings $C_0^\infty \to \mathbb{R}$. As peek-a-boo notes, that's really all there is to say.

---

Half-explanation of why we think of distributions as "generalized functions":

Some distributions can be constructed from, say, continuous functions $f : \mathbb{R} \to \mathbb{R}$: specifically, such an $f$ can be used to construct a distribution $D_f : C_0^\infty \to \mathbb{R}$ via $D_f(\psi) := \int f \psi$. So in some sense, every nice function $f$ has a corresponding distribution $D_f$.

However, there are distributions $D$ that cannot be written in this way (i.e. there exists no $g:\mathbb{R} \to \mathbb{R}$ such that $D(\psi) = \int g \psi$ for all $\psi$). So the collection of distributions contains "more" than nice functions, so it is more "general."