Let $G$ be a locally compact group with Haar measure $\mu$ and let $X$ be a Banach space. Define a norm on $C_c(G,X)$ via $$\|f\|_{1}:=\int_{G}\|f(t)\|_{X} \ \text{d}\mu(t).$$ Also note that $C_c(G)\otimes X$ can be viewed as a linear subspace of $C_c(G,X)$. Indeed, any tensor $a\otimes x\in C_c(G)\otimes X$ acts on $t\in G$ via $(a\otimes x)(t):=a(t)x$.
I am trying to prove that
$C_c(G)\otimes X$ is dense in $C_{c}(G,X)$ w.r.t. $\|\cdot\|_1$.
I found a prove that shows density with respect to the inductive limit topology, and apparently this already implies density w.r.t. $\|\cdot\|_{1}$. This proof uses a partition of unity (which is possible, since $G$ is paracompact).
Is there a more elementary direct proof of this statement without the use of the inductive limit topology?
UPDATES AND PROGRESS:
I think I can reduce the proof to the following claim:
Let $G$ be a locally compact group with identity element $e$. Given $\delta>0$, $f\in C_{c}(G,X)$ and compact neighbourhood $W\ni e$ (i.e. W is compact and contains an open neighbourhood of $e$), there exists an open neighbourhood $V\ni e$ contained in $W$ such that $$r^{-1}s\in V\quad\implies\quad\|f(s)-f(r)\|\leq\delta\quad\quad\forall r,s\in G.$$
Namely, assuming the claim is true, here is what I tried so far:
Let $f\in C_{c}(G,X)$ and $\varepsilon>0$ be given. Let $W\ni e$ be a compact neighbourhood. Choose $V$ as in the claim for $\delta:=\epsilon/\mu(KW)$, where $K:=\text{supp}(f)$. Since $K$ is compact, there exist $r_{1},\ldots,r_{n}\in K$ such that $K\subset\bigcup_{j=1}^{n}r_{j}V$. Then $G\setminus K, r_{1}V,\ldots,r_{n}V$ is an open cover of $G$. Using a partition of unity argument, we find $a_{0},a_{1},\ldots,a_{n}\in C_{c}(G)$ such that
- $\text{supp}(a_{0})\subset G\setminus K$ and $\text{supp}(a_{j})\subset r_{j}V\subset r_{j}W$ for $j=1,\ldots,n$.
- $0\leq\sum_{j=0}^{n}a_{j}\leq1$.
- $\sum_{j=0}^{n}a_{j}(s)=\sum_{j=1}^{n}a_{j}(s)=1$ for all $s\in K=\text{supp}(f)$.
Define $g:=\sum_{j=1}^{n}a_{j}\otimes f(r_{j})\in C_{c}(G)\otimes X$. Then we have \begin{align*} \|f-g\|_{1}&=\int_{G}\bigg\|f(s)-\sum_{j=1}^{n}a_{j}(s)f(r_{j})\bigg\| \ \text{d}\mu(s)\\ &=\int_{G}\bigg\|\sum_{j=1}^{n}a_{j}(s)(f(s)-f(r_{j}))\bigg\| \ \text{d}\mu(s),\quad (\text{by 3.})\\ &\leq\int_{G}\sum_{j=1}^{n}a_{j}(s)\|f(s)-f(r_{j})\| \ \text{d}\mu(s)\\ &\leq\delta\int_{G}\sum_{j=1}^{n}a_{j}(s) \ \text{d}\mu(s),\quad(\text{by 1. and by def. of $V$})\\ &\leq\frac{\epsilon}{\mu(KW)}\mu(KW)=\epsilon, \end{align*} where we used in the last step that $\sum_{j=1}^{n}a_{j}\leq1$ by 2. and $\text{supp}(\sum_{j=1}^{n}a_{j})\subset KV\subset KW$ by 1.
Is this proof correct? And is the claim I use true?