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.


1 Answer 1


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.