Today I was very surprised to learn that $C_c^0(\Omega)$ is not a Banach space with the supremum norm. Why is that, when $C^\infty_c(\Omega)$ is?
Also (from here, bottom of page 18), I learn that even though we can approximate any function in $C_c^0(\Omega)$ by a $C_c^\infty(\Omega)$ function as closely as we want, there is no dense inclusion since the space is not Banach. Is it really necessary for "density" that being complete is required?
If completeness is required for density, then the statement I read somewhere
$C_c^\infty(0,T;V)$ ($V$ is Hilbert) is dense in $W(0,T)$ (some space)
does not make sense either. Or is it different because $[0,T]$ is compact?
It should not be a surprise that $C^0_c(\Omega)$ is not a Banach space. The sequence of functions with compact supports can easily converge to a function which is nonzero on all of $\Omega$. For example, the function $f(x)= \mathrm{dist}\,(x,\partial \Omega) $ does not belong to $C_c^0(\Omega)$, but $f_n(x)=\max(0,f(x)-1/n)$ do, and $f_n\to f$ uniformly.
Concerning density: on the top of page 12 the author defines the concept of a dense subset only for Banach spaces. Therefore, a subset of a $C^0_c(\Omega)$ cannot be dense, simply because the notion of being dense is not defined in this space. This is not a position that I would take, but the author of a book gets to decide what to define and how.