Let \(X\subseteq Y\) be Banach spaces, and assume that there exists a continuation map {\^}: \(X^*\to Y^*\) (where \(X^*\), resp. \(Y^*\) denotes the dual of X, resp. Y) such that
(i) \({\hat \phi}|_ X=\phi\), \(\phi \in X^*,\)
(ii) {\^} is linear, and
(iii) there exists \(c>0\) such that for each \(y\in Y\), for all \(\phi_ 1,...,\phi_ m\in X^*\), for each \(\epsilon >0\) there exists \(x\in X\) such that \(| {\hat \phi}_ i(y)-\phi_ i(x)| <\epsilon\), \(i=1,...,m.\)
Then {\^} is continuous, and there exists a continuous, linear extension operator {\^}: \({\mathcal F}(X)\to {\mathcal F}(Y)\) (here \({\mathcal F}(X)\), resp. \({\mathcal F}(Y)\), denotes the space of the finite-dimensional operators on X, resp. Y); if X has the approximation property then {\^} has a continuation to {\^}: \({\mathcal K}(X)\to {\mathcal K}(Y)\), where \({\mathcal K}\) denotes the space of compact operators. Moreover, the image Im(Ĉ)\(\subseteq X\) for each \(C\in {\mathcal K}(X)\). As an example, the existence of a continuation map {\^} is shown for \(X=C[a,b]\), \(Y=\otimes^{n}_{i=1}C[a_ i,b_ i]\), where \(\{[a_ i,b_ i]|\) \(i=1,2,...,n\}\) is a finite covering of [a,b].