Geodesic
The uniqueness of order in the spaces $L_p[0,1]$ and $l_p$
Matematičeskie zametki, Tome 18 (1975) no. 3, pp. 313-325.

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It is considered to what extent the order in the spaces $L_p$ and $l_p$ is determined by the linearly topological type. It is proved that, for example, $L_1$ and $L_2$ have a unique continuous order, and the spaces $l_p$, $p\ne2,\infty$, admit only discrete orders. 
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     title = {The uniqueness of order in the spaces $L_p[0,1]$ and $l_p$},
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     year = {1975},
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Yu. A. Abramovich; P. Voitaschik. The uniqueness of order in the spaces $L_p[0,1]$ and $l_p$. Matematičeskie zametki, Tome 18 (1975) no. 3, pp. 313-325. http://geodesic.mathdoc.fr/item/MZM_1975_18_3_a0/