$p$-summable series in a Banach space

Question

Let $E$ be a Banach space and denote its dual space by $E^*$. Let $p \in [1, \infty)$ and $x : \mathbb{N}\rightarrow E$ be such that for every $\phi \in E^*$, $$\left( \sum_{n=1}^{\infty} \lvert \phi(x(n))\rvert^p \right)^{1/p}$$ is finite. Why is $$\sup_{\phi \in E^*, \lVert \phi \rVert \leq 1} \left( \sum_{n=1}^{\infty} \lvert \phi(x(n))\rvert^p \right)^{1/p}$$ finite.

Answer 1

This follows directly from the Orlicz–Pettis theorem (applied to the $p^{{\rm th}}$ power of your expression).

You can however avoid using this theorem by showing that the function

$$f(\phi) = \left( \sum_{n=1}^{\infty} \lvert \phi(x(n))\rvert^p \right)^{1/p}\quad (\phi\in B_{E^*})$$

is continuous with respect to the weak*-topology. Then it will be bounded as $B_{X^*}$ is compact in the weak*-topology by the Banach–Alaoglu theorem.