Let $K$ be a locally compact Hausdorff space and $X$ be a Banach space. Denote by $C_0(K,X)$ the Banach space of all continuous $X$-valued functions defined on $K$ that vanish at infinity, equipped with the supremum norm.
($f: K \to X$ vanishes at infinity if for every $\varepsilon > 0$ there exists a compact subset $K_\varepsilon$ of $K$ satisfying $\|f(k)\| < \varepsilon$ for all $k \in K \setminus K_\varepsilon$)
For a compact Hausdorff space $L$, denote $C_0(L,X)$ simply by $C(L,X)$.
Is it true that for every locally compact Hausdorff space $K$ and Banach space $X$ there exists a compact Hausdorff space $L$ such that the spaces $C_0(K,X)$ and $C(L,X)$ are isomorphic?
I only know this is true for the special case when $K$ is an infinite set equipped with the discrete topology (it suffices to define $L$ as the Alexandroff compactification of $K$).