I'm a little embarrassed to be asking this, but surely there is a simple argument that I didn't see?


Let $(f\_\lambda)$ be a net in $l^\infty$ which converges weak\* to $f \in l^\infty$. We do not assume the net is bounded. Does the net $(f\_\lambda^+)$ converge weak\* to $f^+$, where $f^+ = \max(f,0)$ is the positive part of $f$?


It's false in $L^\infty[0,1]$.



Weak* continuity of positive parts
Given a finite set $\cal F$ of functions in $\ell\_1$, choose a function $z\_{\cal F}$ in $\ell\_\infty$ s.t. $\langle z\_{\cal F}, x \rangle =0$ for all $x$ in $\cal F$ s.t. $z\_{\cal F}$ has at least one positive coordinate, and normalized s.t. $\langle z^+\_{\cal F}, u \rangle = 1$, where $u := \sum\_{n=1}^\infty 2^{-n} e\_n$ and $e\_n$ is the $n$th unit vector in $\ell\_1$. The net $(z\_{\cal F})$ converges weak$^\*$ to zero when the finite subsets of $\ell\_1$ are directed by inclusion.


