Some property of Banach space

Question

When $X$ is Banach space and $\{f_i\}\subseteq X^{\ast}$, why for any $x\in X,$ $\sum_{i=1}^{\infty}|f_i(x)|<\infty$ if and only if for any $F\in X^{\ast\ast}$, $\sum_{i=1}^{\infty}|F(f_i)|<\infty$? Any ideas would be appreciated.

Answer 1

Hints: One direction is straightforward.
For the other direction, apply the Banach-Steinhaus theorem for $\displaystyle g_n:=\sum_{i\le n} \epsilon_i\,f_i$ where $\epsilon_i\in\{-1,+1\}$ are fixed for a given $F\in X^{**}$.

Fix an $F\in X^{**}$ and choose $\epsilon_i$ so that $|F(f_i)|=\epsilon_i\cdot F(f_i)$.
For each $x\in X$, we have $\displaystyle |g_n(x)|\le\sum_{i=0}^\infty|f_i(x)|<\infty$, thus by Banach-Steinhaus, they are uniformly bounded: $M:=\sup_n\|g_n\|<\infty$.
Finally, we have $\displaystyle \sum_{i=0}^n |F(f_i)|\,=\,\sum_{i=0}^n \epsilon_i\,F(f_i)\,=\,F(g_n)\,\le\,\|F\|\cdot\|g_n\|\,\le\,\|F\|\cdot M\,\,<\,\infty$.