# Why we should have $|\alpha|\leq N_K$ in the definition of distribution continuity?

In Wiki the definition of T is a continious distribution given by :$$T$$ is continuous distribution if and only if for every compact subset $$K$$ of $$U$$ there exists a positive constant $$C_K$$ and a non-negative integer $$N_K$$ such that:

$$|T(\phi)|\leq C_K\sup{|{\partial} ^{\alpha}\phi(x)| | x\in K,|\alpha|\leq N_K}$$ with $${\alpha}$$ is Multi-index , Now my question here is : Why we should have $$|\alpha|\leq N_K$$ to get a continuous distribution ?

This is exactly a somewhat formulaic description of the topology on test functions $$\mathcal D$$, (a strict colimit of Frechet spaces $$\mathcal D_K$$, test functions supported on compacts $$K$$). The colimit property exactly is that a continuous linear map $$T:\mathcal D\to V$$ for some other (locally convex) topological vector space is given by a (compatible in the obvious way) collection of continuous linear maps $$T_K:\mathcal D_K\to V$$.

Thus, the issue of continuity is exactly about continuity on the Frechet "limitands" $$\mathcal D_K$$. These are (projective) limits of Banach spaces, namely, the completions of $$\mathcal D_K$$ with respect to the norms $$\nu_N(f)=\sup_{x\in K}\sup_{|\alpha|\le N} |f^{(\alpha)}(x)|$$. Why the latter? Because $$C^{(N)}(K)$$ is indeed complete, with this norm. That is, it is the "correct" topology to put on $$C^{(N)}(K)$$, and inevitably entails the (Frechet) limit topology on $$C^\infty(K)$$, etc.