Extension of vector-valued functions and weak\u2013strong principles for differentiable functions of finite order

Abstract

In this paper we study the problem of extending functions with values in a locally convex Hausdorff space E over a field mathbb {K}, which has weak extensions in a weighted Banach space {mathcal {F}}nu (Omega ,mathbb {K}) of scalar-valued functions on a set Omega, to functions in a vector-valued counterpart mathcal {F}nu (Omega ,E) of {mathcal {F}}nu (Omega ,mathbb {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.