I am studying distribution theory. Denoting by $C_c^\infty$ the space of infinitely diferenciable function with compact support. Many authors point out the topology generated by the norms on the space $C_c^\infty(K)$ with $K$ compact (thats is the funcions in $C_c^\infty$ with support in $K$) is not complete to justify the introduction of the machinery of inductive limit topology on $C_c^\infty$ to turn it complete. Well, completeness is certainly usefull, however up to now I did not see where exactly the theory break down, without completness.
Somebody could give me some hints? Maybe it arises in applications...
Sincerely.
Edit - A kind of self answer: Let $C^\infty_c(\mathbb R)$ equipped with the non complete topology, it is the topology corresponding to the uniform conergence (of any derivatives) on the compacts (I am right?). Now, let $\Lambda\in C^\infty_c(\mathbb R)*$ (the algebraic dual), $\Lambda$ is continuous implies $\Lambda$ is continuous on any $C^\infty_c(K)$ but the converse is false. Take $\varphi_n=\xi(x-n)$ with $\xi$ in $C^\infty_c$ with support in $[0,1]$ and $\int \xi =1$. We have $\varphi_n \to 0$ (as well as its derivatives) uniformely on the compact, but
$$\Lambda(\varphi_n) = \int \varphi_n = 1 \neq \Lambda(0)$$
also for any compact $K$ there exists a constant $C>0$ such that:
$$|\Lambda(\varphi)| \leq C\sup_{x\in K} |\varphi|$$
so we have no "good" continuity criterion for the linear form. The inductive limit topology provides the good one since bounded subset belongs to a $C_c^\infty(K)$.
Is I am right!? If it is the case, it is confuse to me, because all the textbook I found, such as the one of Rudin say something like: "this topo is not complete, so let construct an other one which turn it complete" to justify the construction and this messed up. It seems (to me) the problem is more about characterization of continuous linear form on $C^\infty_c(\Omega)$, and for free we get completeness...