Series and absolute convergence in a normed space

Let $V$ be a normed space. A sequence in $V$ is a function $x:\mathbb{N}\to V$, written $(x_n{)}_{n\ge 0}$ with $x_n:=x(n)$ (The natural numbers $\mathbb{N}$ (von Neumann)). Fix such a sequence.

Define its vector partial sums recursively by

$s_0:=0_V,s_{m+1}:=s_m+x_m\mathrm{.}$

We write $s_m={\sum }_{n<m}{x}_n$. The series ${\sum }_{n=0}^{\mathrm{\infty }}{x}_n$ converges when these partial sums converge in $V$. Its sum is then the limit of the partial sums.

The series is absolutely convergent when the scalar series ${\sum }_{n=0}^{\mathrm{\infty }}\mathrm{\parallel }{x}_n\mathrm{\parallel }$ converges in the sense of Series, partial sums, convergence and the sum, divergence, and the tail series.

• Absolute convergence is a statement about the scalar series of norms, not a second notion of convergence in $V$.
• The tail identity $s_m-s_{\mathrm{\ell }}={\sum }_{\mathrm{\ell }\le n<m}{x}_n(\mathrm{\ell }<m)$ is the finite-sum identity used in every norm estimate below.