= Sharp comparison of l1 and l2 norms
{title2=$\|x\|_2\leq\|x\|_1\leq\sqrt n\|x\|_2$}
For $x\in\mathbb R^n$, <Cauchy-Schwarz inequality> applied to $(|x_i|)$ and $(1)$ gives the upper bound. Equal absolute coordinate values attain its factor $\sqrt n$. Expanding $(\sum_i|x_i|)^2$ gives the lower bound, and a vector with only one nonzero coordinate attains its factor one. Thus both constants are optimal. These <Lp norms> define the same finite-dimensional topology despite having different unit balls.
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