# extension of a finite dimensional subspace

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.

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 $S$.