We have known that if $X$ is a Banach space and $\sum_{n=0}^{\infty}x_n$ is an absolutely convergent series in $X$ then $\sum_{n=0}^{\infty}x_n$ is a convergent series. Moreover, we have $$ (*)\quad \left\|\sum_{n=0}^{\infty}x_n\right\|\leq\sum_{n=0}^{\infty}\|x_n\| $$ by the continuity of $\|.\|$ on $X$. My own question is that if we replace $\|.\|$ by a seminorm that is a functional $f$ satisfying
$f(\lambda x)=|\lambda|f(x) \quad \forall \lambda\in\mathbb{R}, \forall x\in X$
$f(x+y)\leq f(x)+f(y) \quad \forall x,y\in X$
thẹn whether the inequality (*) still holds?
My attemption. We can use Zabreiko's lemma (Zabreiko's Lemma) to give the solution for this question by construct a discontinuous seminorm on a Banach space. But I would like to construct explicit Banach space $X$, discontinuous seminorm $p$, and an absolutely convergent series $\sum_{n=0}^{\infty}x_n$ on $X$ satisfying $$ p\left(\sum_{n=0}^{\infty}x_n\right)>\sum_{n=0}^{\infty}p(x_n). $$ Thank you for all helping and comments.