I saw similar quastions about sequences, but this question is about some classes of functions.
Please fill up the gaps to obtain a clean full proof of the following statement: $C_0(R^d)$ is a closed subspace of $L^\infty(R^d)$.
This lemma is important for having proof of Riemann-Lebesgue lemma.
My attempt of proof:
- Goal $C_0(R^d)$ is a subspace of $L^\infty(R^d)$:
1.a) Goal: $C_0(R^d)$ is a subset of $L^\infty(R^d)$. Definition: $C_0(X)=\overline{C_c(X)}^{\|\cdot\|_\infty}$ (Side questions: how from this definition obtain that $\|z_i\|\to\infty$ implies $\|f(z_i)\|\to 0$ for any $f\in C_0(X)$? Is it a criterion of being in $C_0(X)$? I doubt.)
The following statement may be of some use: "Continuous functions are Lebesgue integrable". But it is important to prove that for each $p$, $f$ is in $L^p(R^d)$, so here I am stuck.
I suspect that functions from $C_0(R^d)$ are bounded(?) and continuous(?). It may be of some use. (It would mean that $C_0\subseteq C_b$) ...
1.b) Goal: linear combination of elements of $C_0(R^d)$ stays in $C_0(R^d)$. Linearcombination of Cauchy sequences is a Cauchy sequence, so it stays inside. This is done.
- Goal: $C_0(R^d)$ is closed. This might be proven directly since every element of sequence in $C_0(R^d)$ is represented as sequence in $C_c(X)$: therefore we can take diagonal sequence. Its convergence might follow from all mentioned sequences being Cauchy...? ...
Please fix this unfinished proof details and comments are welcome!
