Do I get it correctly, that the are different definitions of "Banach lattices" available in the literature? To be precise, some authors (like Schaefer) include the order continuity of the norm, while others do not (like Abramovich Aliprantis)?
2026-03-25 15:41:12.1774453272
Different definitions of Banach lattices?
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After small discussion in comments it turns out that you use wrong definition of order-continuity. The standard one is $$ \inf A=0\implies\inf_{u\in A}\Vert u\Vert=0 $$ for each non-empty downwards directed subset $A$ of Banach lattice $X$.
The property you stated in the comment is the definition of Riesz norm. For details see Measure Theory. Vol 3. section 354 by D. H. Fremlin.