Geodesic
Order and norm convergence in Banach lattices
Glasgow mathematical journal, Tome 15 (1974) no. 1, p. 13

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Let(V, ≧, ‖ · ‖) be a Banach lattice, and denote V\{0} by V'. For the definition of a Banach lattice and other undefined terms used below, see Vulikh [4]. Leader [3] shows that, if norm convergence is equivalent to order convergence for sequences in V, then the norm is equivalent to an M-norm. By assuming the equivalence for nets in V we can strengthen this result. 
Wirth, Andrew. Order and norm convergence in Banach lattices. Glasgow mathematical journal, Tome 15 (1974) no. 1, p. 13. doi: 10.1017/S0017089500002032
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[1] 1.Birkhoff, G., Lattice Theory, Amer. Math. Soc. Colloquium Publications 25, 3rd Edition (Providence, R. I., 1967), 366–377. Google Scholar

[2] 2.Kelley, J. L. and Namioka, I., Linear topological spaces (Princeton, 1963), 236–242. Google Scholar | DOI

[3] 3.Leader, S., Sequential convergence in lattice groups, Problems in Analysis (ed. Gunning, R. C.), (Princeton, 1970), 273–290. Google Scholar

[4] 4.Vulikh, B. Z., Introduction to the theory of partially ordered spaces (Groningen, 1967). Google Scholar

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