Proposition 4.21.3. Let $X$ be a LCH space and $E$ be a locally convex space over $K \in \RC$. Identify $C_{0}(X; K) \otimes E$ as a subspace of $C_{0}(X; E)$ under the natural map
then $C_{0}(X; K) \otimes E$ is a dense subspace of $C_{0}(X; E)$.
Proof. Let $\phi \in C_{0}(X; E)$. Using Proposition 4.21.2, assume without loss of generality that $\phi \in C_{c}(X; E)$.
Since $\supp{\phi}$ is compact, so is $\phi(X)$ by Proposition 4.16.2. Let $U \in \cn_{E}^{o}(0)$ be balanced, then there exists $\seqf{y_j}\subset E \setminus \bracs{0}$ such that $\bigcup_{j = 1}^{n} (y_{j} + U) \supset \phi(X)$. For each $1 \le j \le n$, let $V_{j} = \phi^{-1}(y_{j} + U)$, then $\seqf{V_j}$ is an open cover of $\supp{\phi}$ consisting of precompact open sets. By Proposition 4.20.6, there exists a partition of unity $\seqf{\phi_j}\subset C_{c}(X; [0, 1])$ on $\supp{\phi}$ subordinate to $\seqf{V_j}$. For any $x \in E$,
Therefore $(\phi - \sum_{j = 1}^{n} y_{j} \phi_{j})(X) \subset U$.$\square$