Vectors concentrated on one coordinate

Question

Suppose $X$ is a Banach space, $(e_i)$ a normalized basis, $(e_i^*)$ the biorthogonal functionals, and $Y$ a finite codimensional subspace of $X$. Given $N$ and $\varepsilon$, can we find $x\in Y$ with $||x||=1$ and $k>N$ such that $|e_k^*(x)|>1-\varepsilon$? If this is false, is it true if we relax the "arbitrarily far enough" condition, and just want existence of one coordinate? Is this known at least for classical Banach spaces?

Answer 1

Take $X=l^1$ and $Y$ the subspace of sequences with zero sum, $e_k$ standard. Then any $x\in Y$ with $\|x\|=1$ satisfies $|e_k(x)|\leqslant 1/2$ for all $k$.