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In this paper we study the problem of extending functions with values in alocally convex Hausdorff space $E$ over a field $\mathbb{K}$, which have weakextensions in a weighted Banach space $\mathcal{F}\nu(\Omega,\mathbb{K})$ ofscalar-valued functions on a set $\Omega$, to functions in a vector-valuedcounterpart $\mathcal{F}\nu(\Omega,E)$ of $\mathcal{F}\nu(\Omega,\mathbb{K})$.Our findings rely on a description of vector-valued functions as linearcontinuous operators and extend results of Frerick, Jord\'{a} and Wengenroth.As an application we derive weak-strong principles for continuously partiallydifferentiable functions of finite order, vector-valued versions of Blaschke'sconvergence theorem for several spaces and Wolff type descriptions of dualspaces.

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