Extension of vector-valued functions and weak–strong principles for differentiable functions of finite order

In this paper we study the problem of extending functions with values in a locally convex Hausdorff space E over a field K, which has weak extensions in a weighted Banach space Fν(Ω , K) of scalar-valued functions on a set Ω , to functions in a vector-valued counterpart Fν(Ω , E) of Fν(Ω , K). Our findings rely on a description of vector-valued functions as continuous linear operators and extend results of Frerick, Jordá and Wengenroth. As an application we derive weak-strong principles for continuously partially differentiable functions of finite order and vector-valued versions of Blaschke’s convergence theorem for several spaces.