The uniqueness of order in the spaces $L_p[0,1]$ and $l_p$

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.