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I am dealing with $\ell_\infty$ norms, and I need a "mathematical" way to take a vector of $p$ length (We'll call it $x$), and construct a vector pointing in the direction of $\|x\|_\infty$.

I know in English, I can say "And I then construct $\hat x$ such that $\hat x$ is pointing in the direction of the the maximum component of $x$"...... but I'd like a mathematical way to present that.

Thank you all!

EDIT: To be more clear, what I want to happen is all elements of the vector that are NOT equal to the $\ell_\infty$ norm be set to zero.

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I don't understand what you mean by that. What if two components have the same absolute value? –  Qiaochu Yuan Jan 20 '12 at 3:02
What do you mean by pointing in the direction of a positive number, i.e. $(\Vert x\Vert_\infty)$.? –  azarel Jan 20 '12 at 3:29
Question updated to be more explicit –  staticfloat Jan 20 '12 at 7:07

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