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Consider any functional $f$ which does not attain its norm on $L_1[0,1]$ (such $f$ exists by James's theorem, but in this case one can find it without, as an $L_\infty$-function with essential supremum equal to $1$, which is not attained on a set of non-zero measure) and let $Y$ be the kernel of $f$. Let $x\notin Y$, if $\hbox{dist}(x,Y)$ is attained at $y\in Y$, then $f$ would attain its norm on $(x-y)/||x-y||$.