1
$\begingroup$

Let $X$ banach space infinite dimensional. If $v\in X$ such that $v\neq 0$.

I wonder if it is possible to find a closed completion for $S=$span $\{ v\}$ in $X$ ie if is possible to find $H$ subspace closed in $X$ such that $$X=S\oplus H$$

I appreciate any suggestions.

$\endgroup$
2
$\begingroup$

You can use Hahn-Banach to extend $\phi: S\to \mathbb K$, $\lambda v\mapsto \lambda$ to a functional $\Phi: X\to\mathbb K$. Then, $X\to X$, $x\mapsto \Phi(x)v$ is a projector onto $S$.

$\endgroup$

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.