Can Dirac Delta Function Test Functions Be Less Restrictive Than Schwartz Space? I understand that, in $L^2$ Hilbert space, functions do not exist pointwise, so if function $\phi(x) \in L^2$, we cannot speak of a specific value $\phi (0)$, at $x=0$, and so we cannot say $\int_{- \infty}^{\infty} \phi(x) \delta (x) dx = \phi (0)$. 
But if we look at the space of functions, $\psi(x)$ (with $x \in R$), that both exist pointwise (so that $\psi(x)$ has a specifically defined value for any $-\infty \leq x \leq \infty$) and are square integrable from $-\infty$ to $\infty$, then does it make sense to let $\delta (x)$ act on those such that we can say: $\int_{- \infty}^{\infty} \psi(x) \delta (x) dx = \psi (0)$? I know that this simple definition of $\psi(x)$ does not guarantee the existence of the derivative of the delta function in this space but, that not withstanding, I think that $\int_{- \infty}^{\infty} \psi(x) \delta (x) dx = \psi (0)$ would still be true for those $\psi(x)$, especially if we define $$\int_{- \infty}^{\infty} \psi(x) \delta (x) dx \equiv \lim_{n\to\infty} \int_{- \infty}^{\infty} \psi(x) \delta_n (x) dx$$ with something like $\delta_n (x) \equiv \sqrt{\frac{n}{\pi}} \exp (-nx^2)$
What I am getting at is that it seems to me that the space of test functions for $\delta (x)$ need not be something as restrictive as Schwartz (S) or Kantorovich (K) space.
 A: Your example fails; consider $\psi$ given by
$$ \psi(x) = \begin{cases} 1 & x = 0 \\ 0 & x \neq 0 \end{cases} $$
$\psi$ is square integrable, 
$$ \psi(0) \neq 0 = \lim_{n \to \infty} \int_{-\infty}^{\infty} \psi(x) \delta_n(x) \, \mathrm{d} x $$
so in the dual space to square-integrable functions, $\delta \neq \lim_{n \to \infty} \delta_n$.

The most reasonable general space to find delta distributions is, in my opinion, in the dual space to the space of  continuous functions, if you simply want them to exist.

But you should pay attention to the fact that, usually, the point isn't simply to define delta distributions — instead one wants to define a space that:


*

*Contains the objects of interest

*Admits all of the sorts of calculations we want in our application

*Has good analytical properties


The second point is very important. For example, in physics, Fourier transforms are very important, which makes the space of tempered distributions (on Schwartz functions) a popular one to work within. 
