Representing certain vector-valued function spaces as tensor products

Let E be a Banach space. For a topological space X, let \({{\cal C}_b}(X,E)\) be the space of all bounded continuous E-valued functions on X, and let \({{\cal C}_K}(X,E)\) be the subspace of \({{\cal C}_b}(X,E)\) consisting of all functions having a pre-compact image in E. We show that \({{\cal C}_K}(X,E)\) is isometrically isomorphic to the injective tensor product \({{\cal C}_b}(X){{\hat \otimes}_\varepsilon}E\), and that \({{\cal C}_b}(X,E) = {{\cal C}_b}(X){{\hat \otimes}_\varepsilon}E\) if and only if E is finite dimensional. Next, we consider the space Lip(X, E) of E-valued Lipschitz operators on a metric space (X, d) and its subspace Lip_K(X, E) of Lipschitz compact operators. Utilizing the results on \({{\cal C}_b}(X,E)\), we prove that Lip_K(X, E) is isometrically isomorphic to a tensor product \({\rm{Lip}}(X){{\hat \otimes}_\alpha}E\), and that \({\rm{Lip}}(X,E) = {\rm{Lip}}(X){{\hat \otimes}_\alpha}E\) if and only if E is finite dimensional. Finally, we consider the space D¹(X, E) of continuously differentiable functions on a perfect compact plane set X and show that, under certain conditions, D¹(X, E) is isometrically isomorphic to a tensor product \({D^1}(X){\hat \otimes _\beta}E\).