On sequence space representation and extension of vector-valued functions

We give a unified approach to handle the problem of extending functions with values in locally convex Hausdorff space $E$ over field $\mathbb{K}$, which have weak extensions $\mathcal{F}(\Omega,\mathbb{K})$ scalar-valued on set $\Omega$, vector-valued counterpart $\mathcal{F}(\Omega,E)$ $\mathcal{F}(\Omega,\mathbb{K})$. The results obtained are based upon representation as linear continuous operators and extend Bonet, Frerick, Gramsch Jordá. In particular, we apply them obtain sequence from known