I'm reading the book "Functions of bounded variation and free discontinuity problem", by L. Ambrosio, N. Fusco and D. Pallara. At page 20, they introduce the space of continuous functions with compact support and its completion
I can't understand the underlined lines: why the say that $C_c(X)$ "turns to be a complete topological vector space"? As far I know $C_c$ is not complete and in fact they define $C_0$ its completion. Moreover they use $C_0(A_h)$, but the space $C_0$ is defined below. Someone can help me? Thanks.
The topology you are thinking of is different from the one they are considering. In your topology $u_k \to u$ if $u_k(x) \to u(x)$ uniformly on compact subsets. With this topology the space is not complete and its completion is $C_0$. But they are considering a topology such that $u_k \to u$ if the supports of the functions $u_k$ are all contained one fixed compact set and $u_k(x) \to u(x)$ uniformly. In this topology the space is complete.