Convergence in space of test functions

Question

In the book Distribution theory by Duistermmat/Kolk convergence in the space of test functions is defined as:

$\phi_j \to \phi$ in $C_0^{\infty}(\Omega) $ (where $\Omega$ is an open subset of $\mathbb{R}^n$ )if:

$i)$ There is a compact set $K: \text{supp} \phi_j \subset K$ for every $j$.

$ ii)$ For every multi-index $\alpha$ the sequence $\partial^\alpha \phi_j \to \partial \phi$ uniformly.

I'm not really sure how to interpret this. By "uniformly" in criterion $ii)$ is it meant uniformly on every compact subset of $\Omega$?

It is not a criterion that for every multi-index $\alpha$ there is a compact $K \subset \Omega$ such that $\text{supp} \partial^\alpha \phi_j$? If no is it possible to $\phi_j \to \phi$ in $C_0^{\infty}(\Omega)$ but the support of its derivatives "diverging"?

Answer 1

(ii) requires "uniformly on $\Omega$", i.e., in the sup norm. (Though it is enough to have "uniformly on $K$ and pointwise on $\Omega\setminus K$", as then $\phi\equiv0\equiv\phi_j$ on $\Omega\setminus K$.)

A. It follows that $\phi(x)=\lim_j \phi_j(x)=0$ outside $K$; hence $\mathrm {supp}\, \phi\subset K$.

B. If you assume $\mathrm {supp}\, \phi\subset K$, then "uniformly in $K$" suffices in (ii).

C. It also follows that $\mathrm {supp}\, \partial^\alpha \phi_j\subset K$ $\forall \alpha\in\mathbb N^n$, $j\in\mathbb N$.

Proof: Let $f\in C^\infty_0(\Omega)$. Assume $\mathrm {supp}\, f\subset K$. As $f\equiv0$ on $K^c$, we have $\partial^\alpha f\equiv0$ on $K^c$, so $\mathrm {supp}\, \partial^\alpha f\subset K$, $\forall \alpha\in\mathbb N^n$, QED.

N.B. As $\phi_j\to\phi$ iff $\phi_j-\phi\to0$, we have $\phi_j\to\phi$ iff

$i')$ There is a compact set $K: \text{supp} (\phi_j-\phi) \subset K$ for every $j$.

$ ii)$ For every multi-index $\alpha$ the sequence $\partial^\alpha (\phi_j-\phi) \to 0$ uniformly on $\Omega$.

Although $ii)$ is clearly equivalent to your $ii)$, condition $i')$ is formally different. However, then $\text{supp}\, \phi_j\subset K\cup K'$, where $K':=\text{supp}\, \phi$, so these two common definitions are clearly equivalent (the union of two compact sets is clearly compact).

N.B. Using either definition, it suffices to assume $\phi\in C^\infty(\Omega)$, by "A." (or that $\phi$ has weak derivatives of all orders, as (ii') guarantees that they are actual derivatives).