= Extreme points of real sequence-space unit balls
{title2=$\operatorname{ext}B_{\ell^\infty}=\{-1,1\}^{\mathbb N},\quad\operatorname{ext}B_{\ell^1}=\{\pm e_j\},\quad\operatorname{ext}B_{\ell^2}=S_{\ell^2}$}
For the real <l-infinity sequence space>, every coordinate of an <extreme point> must be an endpoint of $[-1,1]$; any slack coordinate permits opposite perturbations. For the real <absolutely summable sequence space>, a unit vector with two nonzero coordinates permits a transfer of mass between them, so only signed coordinate vectors are extreme. For the real <l2 sequence space>, the <parallelogram law> makes every unit vector extreme. In all three cases interior points of the unit ball are nonextreme.
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