I'm currently learning distributions. My current intuition is that a distribution is a kind of weak limit in the space of linear functionals $C_c^\infty(U) \rightarrow \mathbb R$ of ordinary functions.
Is this a good intuition? The thing is that when we define certain operations with distributions, for example
- multiplication of two distributions with disjoint singular support
- convolution of two distributions
the operation might not actually commute with weak convergence of sequences in $\mathcal D'$.
For example, let $u$ be the distribution taking $\phi \in C_c^\infty(\mathbb R)$ to $\phi'(0)$. Then note $\langle u, 0\rangle \ne \lim_n\langle u, \sin (nx)\rangle$, as the limit on the right does not exist.
I'm just wondering if there's any way I should think about this. Sorry, I know it's vague.