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10:13 AM

On MSE there are tags called banach-lattices and vector-lattices (with riesz-spaces as a synonym.) Would perhaps some of these tags be useful also on MO?

There was recently this question:

Q: A question on order unbounded sequences in Banach lattices

Let $E$ be a Banach lattice. It is well-known that every norm convergent sequence in $E$ admits an order convergent subsequence and hence admits an order bounded subsequence. But it seems that a norm convergent sequence in $E$ needs not be order bounded. Is there any counterexample? Thank you!