Just a simple question, I am wondering since the test function $\varphi$ is smooth and has compact support, it is in the domain of $D$(Rn), but is it contained in the set of all distributions denoted by $D'$(Rn) (which is a vector space)? From my understanding (possibly wrong) not all distributions have compact support, thankyou!
*the n in Rn should be superscripted
**if possible, since I mentioned $D'$(Rn), does the dual pairing imply an inner product in the vector space $D'$(Rn)? or does the dual pairing equal the inner product on the space of all set of distributions?
The space of test functions on the domain $\Omega$ is denoted by $\mathcal{D}(\Omega)$ or $C_c^\infty(\Omega)$. The latter notation says more about what they are: infinitely differentiable functions ($C^\infty$) with compact support (${}_c$).
Every continuous functions $f$ also generates a distribution $u_f$ by $$\langle u_f, \varphi \rangle := \int f(x) \, \varphi(x) \, dx.$$
Therefore, $C^\infty(\Omega)$ and its subspaces, e.g. $C_c^\infty(\Omega)$, are considered as subspaces of $\mathcal{D}'(\Omega)$.
The pairing $\langle u, \varphi \rangle$ does not imply an inner product on $\mathcal{D}'(\Omega)$.