# A Hahn-Banach type extension problem for multiple functionals

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.

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 $$\lambda_1=\dotsc=\lambda_n=0$$).