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Let $(X,\|\cdot\|)$ be a normed linear space. Recall from prior results that $(X,\|\cdot\|)$ is Banach $\iff$ any absolutely convergent series in $(X,\|\cdot\|)$ converges.
(a) Give an example of a Banach space $(X,\|\cdot\|)$ and a convergent series which is not absolutely convergent.
(b) Give an example of a normed linear space $(X,\|\cdot\|)$ and an absolutely convergent series which is not convergent.
I haven't tried much, I really don't know where to begin.
Let's start with (a), the simplest (and most known) example is the Banach space $\mathbf R$ with the absolute value as norm, and the series $$ \sum_n (-1)^n \frac 1n $$ which converges by the Leibnitz criterion, but is not absolute convergent.
For (b), let $X = c_{00}$, the real seqences with only finitely non-zero terms with the supnorm. Denote by $e_n \in c_{00}$ the sequences constisting only of zeros but a one as the $n$-th term. Consider the sequence $$ \sum_n \frac 1{n^2} e_n $$ which converges to $(n^{-2})$ in $c_0$ (but this is not an element of $c_{00})$, so it does not converge in $c_{00}$, but converges absolutely, as $\sum_n n^{-2} < \infty$:
