Series in a Real Inner Product Space

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $E$ be a real inner product space, with its norm $|\cdot|$ and distance $d$, and let $(x_{k})_{k\in\mathbb{N}}$ be a sequence in $E$.

1. (Partial sums)(i) For $n\in\mathbb{N}$, the $n$-th partial sum of $(x_{k})$ is the finite sum in $E$

$$s_{n}=\sum_{k=1}^{n}x_{k}$$

of the restriction of $(x_{k})$ to $[n]$, which by claim 1 of Properties of Finite Sums of Vectors does not depend on which map extending those summands is used; $(s_{n})_{n\in\mathbb{N}}$ is a sequence in $E$, called the sequence of partial sums of $(x_{k})$.

2. (Convergence and sum)(ii) The series $\sum_{k=1}^{\infty}x_{k}$ converges if the sequence of partial sums converges in $(E,d)$, and diverges otherwise. When it converges, the limit of the sequence of partial sums is unique by Uniqueness of Limits in a Metric Space; that limit is the sum of the series, and is also written $\sum_{k=1}^{\infty}x_{k}$.

3. (Absolute convergence)(iii) The series $\sum_{k=1}^{\infty}x_{k}$ converges absolutely if the series of real numbers $\sum_{k=1}^{\infty}|x_{k}|$ converges.