I am (roughly) quoting from exercise 2.6 in "Topics in Banach Space Theory"
"Suppose $X$ is Banach with separable dual. If $\sum x_n^*$ is a series in $X^*$ such that every subseries converges weak-*, then $\sum x_n$ converges in norm."
How is $x_n^*$ supposed to be interpreted here? Does it mean any biothorgonal functionals?
Doesn't this imply that if $\sum x_n^*$ converges weak-*, then $\sum x_n$ converges in norm?
This appears to just be a typo, and it should say $\sum x_n^*$ at the end instead of $\sum x_n$. That is, $(x_n^*)$ is just an arbitrary sequence in $X^*$, and the ${}^*$ has no special meaning.