Isometries between completely regular vector-valued function spaces

In this paper, first we study surjective isometries (not necessarily linear) between completely regular subspaces \(A\) and \(B\) of \(C_0(X,E)\) and \(C_0(Y,F)\) where \(X\) and \(Y\) are locally compact Hausdorff spaces and \(E\) and \(F\) are normed spaces, not assumed to be neither strictly convex nor complete. We show that for a class of normed spaces \(F\) satisfying a new defined property related to their \(T\)-sets, such an isometry is a (generalized) weighted composition operator up to a translation. Then we apply the result to study surjective isometries between \(A\) and \(B\) whenever \(A\) and \(B\) are equipped with certain norms rather than the supremum norm. Our results unify and generalize some recent results in this context.