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?

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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 $k$.