Let $X$ be a closed subspace of a Banach space Y. I have functionals $f\_0, f\_1, \ldots, f\_n\in X^\*$ such that $f\_0$ is in the span of the remaining ones. I fix an extension of $f\_0$ to $Y$; let me call it $F\_0$. Can I extend them to functionals $F\_1, \ldots, F\_n$ on $Y$ in a way that $F\_0$ is in the span of $F\_1, \ldots, F\_n$? I don't really care about preserving the norms of the original functionals.



A Hahn-Banach type extension problem for multiple functionals
If $f\_0\ne0$ or if $f\_0=0$ and the $f\_1,\dotsc,f\_n$ are not linearly independent, then the answer is trivial:


In this case there is another functional, say $f\_1$, which is in the span of the remaining ones, say $f\_1=\sum\_{k\ne1}\lambda\_kf\_k$ with $\lambda\_0\ne0$: Extend the $f\_k$ to $F\_k$ $(k=2,\dotsc,n)$ and put $F\_1=\sum\_{k\ne1}\lambda\_kF\_k$.


In the remaining case, $f\_0=0$ and $f\_1,\dotsc,f\_n$ being linearly independent, the answer to your question is obviously positive if and only if $F\_0=0$ (because $\sum\_{k=1}^n\lambda\_kF\_k=F\_0$ implies by restriction to the subspace $\lambda\_1=\dotsc=\lambda\_n=0$).


