Example of a linear functional, but not a distribution

Question

I'm looking for an example of a linear functional $u: C_c^\infty(\Omega) \to \mathbb C$ ($\Omega \subset \mathbb R^n$ open), which is not a distribution. I could not find anything... I thought of something like taking the function $\frac 1 x$ and defining $u$ as $$\langle u, \phi \rangle = \int_{\Omega} \frac 1 x \phi(x) \, \mathrm dx \quad \text{for } \phi \in C_c^\infty(\mathbb R) \; ,$$ but this does not seem to work, because the integral is not defined for all $\phi \in C_c^\infty(\mathbb R)$. This is also the reason why we introduce the principle value of $\frac 1 x$. Does anybody know an example?

Answer 1

You can concoct an example using a Hamel basis like in Discontinuous linear functional. Maybe this are the only possible examples like in the case of Banach spaces.