Suppose $x$ is a series in $\ell_1(L_p(\mu))$, which is the space of unconditionally convergent series in the Banach space $L_p(\mu)$. I read that a norm can be defined on this space as
$$w_1(x)=\sup\limits_{||x'||\leq 1} \sum_{k=1}^{\infty} |x'(x(k))|$$
Why is this a well-defined norm, specifically why is $w_1(x)<\infty$ for every series as above?
Recall that for scalar series, unconditional convegrence is equivalent to absolute convergence. For every $x'$, the series $\sum x'(x(k))$ is unconditionally convergent series, hence $\sum |x'(x(k))| <\infty$.
It remains to promote this to a uniform bound. Naturally, the tool to use is the uniform boundedness principle. For each $N$, the truncated sequence $(x(1),\dots, x(N), 0, 0, \dots)$ defines a linear operator $T_N$ from the dual space of $L_p(\mu)$ into $\ell^1$, namely $$ x'\mapsto (x'(x(1)), \dots, x'(x(N)), 0, \dots) $$ By the first paragraph, the family $\{T_N\}$ is pointwise bounded, meaning that for every $x'$ there is a uniform bound on $\|T_N(x')\|_1 $ independent of $N$; indeed this bound is $\sum |x'(x(k))|$. The UBP implies there is $M$ such that $\|T_N\|\le M$ for all $N$. Hence, $$ \sum |x'(x(k))| \le M\|x'\| $$ for all $x'$.