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 $\mathbb{K}$, which have 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 linear continuous operators and extend results of Frerick, Jord\'{a} and Wengenroth. As an application we derive weak-strong principles for continuously partially differentiable functions of finite order, vector-valued versions of Blaschke's convergence theorem for several spaces and Wolff type descriptions of dual spaces.