  We give a unified approach to handle 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 space $\mathcal{F}(\Omega,\mathbb{K})$ of
scalar-valued functions on a set $\Omega$, to functions in a vector-valued
counterpart $\mathcal{F}(\Omega,E)$ of $\mathcal{F}(\Omega,\mathbb{K})$. The
results obtained base upon a representation of vector-valued functions as
linear continuous operators and extend results of Bonet, Frerick, Gramsch and
Jord\'{a}. In particular, we apply them to obtain a sequence space
representation of $\mathcal{F}(\Omega,E)$ from a known representation of
$\mathcal{F}(\Omega,\mathbb{K})$.
