I have an elementary question about finite-dimensional subspaces and finite-codimensional subspaces. This question may be known.

Question. Let $U$ be a finite-dimensional subspace of an infinite-dimensional Banach space $X$. Let $\epsilon>0$. Is there a finite-codimensional subspace $V$ of $X$ such that $U\cap V=\{0\}$ and $$\|u\|\leq (1+\epsilon)\|u+v\|$$ for every $u\in U, v\in V$ ?