Absolutely Summable Series

Definition: Let $(X, \| \cdot \|)$ be a normed linear space. A series $\displaystyle{\sum_{i=1}^{\infty} x_i}$ is said to Converge in $X$ if the sequence of partial sums$(s_n) = \left ( \sum_{i=1}^{n} x_i \right )$ converges in $X$.

We now define absolutely summable series in a normed linear space.

Definition: Let $(X, \| \cdot \|)$ be a normed linear space. A series $\displaystyle{\sum_{i=1}^{\infty} x_i}$ in $X$ is said to be Absolutely Summable if the numerical series $\displaystyle{\sum_{i=1}^{\infty} \| x_i \|}$ converges.

It is important to note that if we consider the $\mathbb{R}$ with the Euclidean norm, then absolute summability and absolute convergence is equivalent. In $\mathbb{R}$, every absolutely convergent series converges in $\mathbb{R}$. However, in a general normed space $(X, \| \cdot \|)$, absolute summability does NOT imply convergence of a series in $X$.