I am studying the concept of distribution and struggle with some examples. A distribution $T \in D'(\Omega)$ on $\Omega$ is a continuous linear map $T:D(\Omega)\to \mathbb{R}$ where $D(U)=C_c^{\infty}(\Omega)$.
Prove or disprove whether the following examples are distributions
a) $\Omega=(0,1),T\phi=\sum_{n=2}^{\infty} \phi^{(n)}(\frac1n)$
b) $\Omega=\mathbb{R},T\phi=\sum_{n=1}^{\infty}\phi^{(n)}(\frac1n)$
c) $\Omega=\mathbb{R}^2, T\phi=\int_0^{2\pi} \phi(\cos(\alpha),\sin(\alpha))d\alpha$,
where $\phi\in D(\Omega)$.
I struggle to even start. $T$ is linear in all cases obviously. I either have to prove $\lim_{k\to\infty} T(\phi_k(x))=T(\lim_{k\to \infty} \phi_k)$ or find a test function for which continuity does not hold. First I guess I need to check the expressions are well-defined. I was thinking because of the $1/n$ term maybe the sum in a)/b) can become infinite for some test function t. But there is no test function satisfying $\phi^{(n)}(\frac1n)=1/n$ so this probably doesn't work.
The main theorem is that locally distributions have finite order. Assume it is not, it means that for $\phi\in C^\infty_c([a,b]),\sum_{m\le k}\|\phi^{(m)}\|_\infty$ small doesn't imply $|S(\phi)|$ small, for each $k$, taking $\phi_k\in C^\infty_c([a,b])$ with $\sum_{m\le k}\|\phi_k^{(m)}\|_\infty< 2^{-k}$ and $S(\phi_k)=1$ we get a sequence $\Phi_K=\sum_{k\le K} c_k \phi_k$ which converges in $C^\infty_c([a,b])$ but such that $\lim_{K\to \infty} S(\Phi_K)=\infty$ contradicting that $S$ is a distribution.
If $T_b$ is a distribution, since it is compactly supported then it has finite order $k$, thus it is the $k+1$-th derivative of a continuous function, which is obviously wrong.
$T_a$ is a distribution on $(0,1)$ because for $\phi\in C^\infty_c([\epsilon,1-\epsilon])$, $T_a(\phi)$ is a finite sum.