Suppose X is a Banach space and $x_1,...,x_n$ are linearly independent vectors. Can we find some sort of lower bound for $||\sum x_i||$? What about if we restict ourselves to normalised vectors?
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Consider $x_1 = e_1, x_2 = (-e_1 + te_2) / \sqrt{1 + t^2}$ in $\mathbb R^2$. – user251257 Mar 09 '16 at 03:49
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Linear independence does not do anything of consequence here, it's not a quantitative property. – Mar 09 '16 at 04:05
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@Sally Of course, it does. Choose $x_2 = -x_1$ in $\mathbb R^2$. – Friedrich Philipp Mar 09 '16 at 04:08
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@Tim In user251257's example, let $t\to 0$. – Friedrich Philipp Mar 09 '16 at 04:09
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@FriedrichPhilipp don't spoil it :D – user251257 Mar 09 '16 at 04:09