Bound in reflexive Banach space + monotonicity of sequence implies weak convergence of sequence? Let $V$ be a reflexive Banach space with a partial order relation $\leq.$ Furthermore, suppose it is a lattice. What further conditions does one need to have this property:

if $v_n$ is a bounded sequence in $V$ and $v_n \leq v_{n+1}$ for all $n$, then $v_n \rightharpoonup v$ in $V$.

Note the weak convergence is asked for for the original sequence (not a subsequence). Does anyone have a reference?

Lets ask for the following compatibility conditions:

*

*If a monotone net $x_\alpha$ has least upper bound $x$ (ie $x_\alpha$ converges in order to $x$) then $x_\alpha$ converges weakly to $x$.

*If a monotone net $x_\alpha$ converges weakly to $x$ then $x$ is the least upper bound of $x_\alpha$.

For example on $L^2(\Bbb R)$ the usual relation $f≤g$ satisfies these two conditions.
Then if you have a monotone bounded sequence $v_n$ you note that it has a limit point $v$ by Banach Alaoglu. By Eberlein Smulian you get a convergent sub-sequence $v_{n_k}$. By condition (2) the weak limit $v$ is a least upper bound of $v_{n_k}$. But the least upper bound of $v_{n_k}$ is a least upper bound of $v_n$ since $v_{n_k}$ is final in $v_n$. Hence $v$ is the least upper bound of the monotone sequence $v_n$ and by (1) $v_n$ converges to $v$.
