On embeddings of \(C_0(K)\) spaces into \(C_0(L,X)\) spaces (Q2804305)

Let \(X\) be a Banach space and \(K\) a locally compact Hausdorff space. Let \(C_{0}(K, X)\) stand for the Banach space of all continuous \(X\)-valued functions on \(K\) which vanish at infinity, provided with the supremum norm. If \(X\) is the scalar field, this space is denoted by \(C_{0}(K)\). The main result of the paper under review states that if \(X\) contains no subspace isomorphic to \(c_{0}\) and \(C_{0}(K)\) is isomorphic to a subspace of \(C_{0}(S, X)\), then either \(K\) is finite or the cardinality of \(K\) is less than or equal to the cardinality of \(S\). This theorem was also obtained independently in [\textit{E. M. Galego} and \textit{M. A. Rincón-Villamizar}, Bull. Sci. Math. 139, No. 8, 880--891 (2015; Zbl 1335.46005)].