section_path: Generalizations/Summations over arbitrary index sets/Series in topological vector spaces/Series in Banach and seminormed spaces
section_name: Series in Banach and seminormed spaces
paragraph_text: Series in Banach and seminormed spaces
  
  The notion of series can be easily extended to the case of a seminormed space. If x n {\displaystyle x_{n}} is a sequence of elements of a normed space X {\displaystyle X} and if x ∈ X {\displaystyle x\in X} then the series ∑ x n {\displaystyle \sum x_{n}} converges to x {\displaystyle x} in X {\displaystyle X} if the sequence of partial sums of the series (∑ n = 0 N x n) N = 1 ∞ {\textstyle \left(\sum _{n=0}^{N}x_{n}\right)_{N=1}^{\infty }} converges to x {\displaystyle x} in X {\displaystyle X}; to wit,
  
  ‖ x − ∑ n = 0 N x n ‖ → 0 as N → ∞. {\displaystyle \left\|x-\sum _{n=0}^{N}x_{n}\right\|\to 0\quad {\text{ as }}N\to \infty.}
  
  More generally, convergence of series can be defined in any abelian Hausdorff topological group. Specifically, in this case, ∑ x n {\displaystyle \sum x_{n}} converges to x {\displaystyle x} if the sequence of partial sums converges to x. {\displaystyle x.}
  
  If (X, | ⋅ |) {\displaystyle (X,|\cdot |)} is a seminormed space, then the notion of absolute convergence becomes: A series ∑ i ∈ I x i {\textstyle \sum _{i\in I}x_{i}} of vectors in X {\displaystyle X} converges absolutely if
  
  ∑ i ∈ I | x i | < + ∞ {\displaystyle \sum _{i\in I}\left|x_{i}\right|<+\infty }
  
  in which case all but at most countably many of the values | x i | {\displaystyle \left|x_{i}\right|} are necessarily zero.
  
  If a countable series of vectors in a Banach space converges absolutely then it converges unconditionally, but the converse only holds in finite-dimensional Banach spaces (theorem of Dvoretzky & Rogers (1950)).