Banach-Stone theorems for vector valued functions on completely regular spaces
spaces of vector-valued continuous functions
This paper is an interesting contribution to the area of functional analysis that deals with the variations on the theme of the classical Banach-Stone theorem. For realcompact topological spaces \(X,Y\) and for locally convex spaces \(E,F\), and for subspaces \(A \subset C(X,E)\), \(B \subset C(Y,X)\), let \(\phi: A \rightarrow B\) be a linear bijection such that \(z(f) \subset z(g)\) if and only if \(z(\phi(f)) \subset z(\phi(g))\) (\(z(f)\) stands for the zero-set). Then a natural question to ask is whether \(\phi(f)(y) = J_y(f(\tau(y))\) for all \(y \in Y\), \(f \in A\), where \(\tau:Y \rightarrow X\) is a homeomorphism and \(J_y:E \rightarrow F\) is a linear bijection.NEWLINENEWLINEThe authors give a positive answer for the entire space of continuous functions and for certain special subspaces. When the domain spaces are metric and range spaces are normed linear spaces, spaces of uniformly continuous or Lipschitz continuous functions fall in this category. Under the assumption that \(\phi\) is continuous with respect to the topology of uniform convergence, one also gets that the family of fiber linear maps \(\{J_y\}\) is equicontinuous.
scientific article; zbMATH DE number 6064301
scientific article