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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.

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impartialmale
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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 $$p(v)>\sum_{k=1}^{\infty} p(v_k) =0.$$

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    Discontinuous functionals on Banach spaces are always non-constructive. OP asks for explicit example but I doubt if it exists. – Norbert Aug 24 '14 at 13:57
  • @Norbert: Here, FisiaiLusia used discontinuous functional on Banach space to construct discontinuous seminorm. However, we can use another way to construct explicitly discontinuous seminorm on Banach space. – impartialmale Aug 24 '14 at 22:04
  • @FisisaiLusia: Thank you for your solution. – impartialmale Aug 28 '14 at 21:41