Discontinuous seminorm on Banach space 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.

There exists a discontinuous functional $f:X\to \mathbb{R}$ take $p(x) =|f(x)|$ . Since $f$ is discontinuous then $\mbox{Ker}(f)$ is dense in $X.$ Take any $v\in X\setminus \mbox{Ker}(f)$ and construct absolutely convergent (with respect to norm) series such that $v=\sum_{k=1}^{\infty} v_k$ then you have $$p(v)>\sum_{k=1}^{\infty} p(v_k) =0.$$
