Understanding denseness of $C^\infty$ in $L^p$ space.

Solution 1:

By looking at the proof, I guess there are two points that can be confusing. Feel free to ask more questions if this does not answer you original one.

The idea, as Martin and Rasmus suggested, is to take an arbitrary function $f\in L^p$, and for all $\epsilon>0$, find a smooth function $g$ with compact support such that $\|f-g\|_p<\epsilon$. In this proof, we have $g=(f\chi_K)*\phi_t$, where $\phi$ is a $C^\infty$-function with compact support and integral equal to $1$, and $K$ is a suitably chosen compact set. Now, in the proof, we have the following statement:

Let $K$ be a compact set chosen in such a way that $\|f\chi_{K^c}\|_p<\epsilon/2$.

This can be done, because $f\in L^p$ (so finite $L^p$-norm) and because $\mathbb{R}^n$ is a countable union of compact sets.

Finally, when choosing $t_0$, we use Theorem 1.4, and the fact that $(f\chi_K)*\phi_t$ is a $C^\infty$-function with compact support for all $t>0$ follows from Proposition 1.2 and Theorem 1.3.