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Abstract

For a locally compact Hausdorff space $K$ and a Banach space $X$ let $C_0(K, X)$ denote the space of all continuous functions $f:K\to X$ which vanish at infinity, equipped with the supremum norm. If $X$ is the scalar field, we denote $C_0(K, X)$ simply by $C_0(K)$. We prove that for locally compact Hausdorff spaces $K$ and $L$ and for a Banach space $X$ containing no copy of $c_0$, if there is an isomorphic embedding of $C_0(K)$ into $C_0(L,X)$, then either $K$ is finite or $|K|\leq |L|$. As a consequence, if there is an isomorphic embedding of $C_0(K)$ into $C_0(L,X)$ where $X$ contains no copy of $c_0$ and $L$ is scattered, then $K$ must be scattered.