On the distribution space ${\cal E'}(\mathbb R^n)$ $${\cal D}'(\mathbb R^n) \equiv \{ {\rm continuous \ linear \ functionals } \ \ u_f(\phi): {\cal D}(\mathbb R^n) \to \mathbb C, \ \  {\rm with }\  f \in L^1_{loc}(\mathbb R^n) \ {\rm and} \ \phi \in {\cal C}^{\infty}_c (\mathbb R^n)\}$$
I'd like to write a similar definition for ${\cal E'}(\mathbb R^n)$, if it's possible.
Attempt.
$${\cal E}'(\mathbb R^n) \equiv \{ {\rm continuous \ linear \ functionals } \ \ v_g(\psi): {\cal E}(\mathbb R^n) \to \mathbb C, \ \  {\rm with }\  g \in {\cal C}^\infty_c(\mathbb R^n) \ {\rm and} \ \psi \in {\cal C}^{\infty} (\mathbb R^n)\}$$
Is this correct? Or better, is it equivalent to the usual definitions for $\cal E'(\mathbb R^n)?$
 A: I think one disserves oneself by thinking that "functions" and "distributions" are fundamentally different. Yes, many formalizations would seem to make them wildly different things from each other... but the historical and on-going motivations certainly do not make them wildly different.
In particular, in practice, "distributions" appear as ("weak" or other non-pointwise...) limits of very nice functions, for example, test functions.
So, while uniformly-on-compacts pointwise limits of continuous functions certainly are continuous, other limits (like $L^2$, without even mentioning "distributions") of continuous functions need not be continuous.
Also, a point that only dawned on me a few years ago is that probably no physicist every really thought that the "potential" $\delta$ really had $0$ extent but infinite value... but, rather, wrote $\delta$ for an idealization of a potential with very small extent but total mass (or whatever) $1$. So the fact that this is classically mathematically impossible was not too relevant, since the manipulations were really partly a narrative about physical-mathematical reasoning... and possibly not at all pretending to assert anything about a literal $\delta$.
And, again, indeed, most topological vector spaces have no natural imbedding into their duals.
But a/the fundamental interest in the space of distributions is that the space of test functions does imbed in it, and is dense. Yes, this is by the identification of a test function with the integrate-against-it functional, ... which turns out to be a smart choice in practice! As opposed to thinking about functions as things that have inputs numbers and outputs numbers. Further, among distributions, the compactly-supported ones are (provably) finite linear combinations of some (finite-order) derivatives of compactly-supported continuous functions. So we've not added superfluous things, if we reeeeeally want to be confident of differentiating anything at all.
Things become a little disordered when we look at the Frechet space of smooth functions, and its dual, the compactly-supported distributions. It requires a little work to prove that the dual is indeed what it is. And the space of smooth functions does not imbed in that dual in any natural way. Too bad. Still, test functions are dense in it (imbedding by the integrate-against convention).
