---
tag: XYTN19EPF2
kind: concept
name: Sequence space
msc: [46A45, 40H05]
sources:
  - url: https://en.wikipedia.org/wiki/Sequence_space
same_as:
  wikidata: Q289411
  wikipedia: Sequence space
---

## Summary

A sequence space is a vector space whose elements are infinite sequences of real or complex numbers, with termwise addition and scalar multiplication.
Standard examples are the [[spaces ℓ^p|08P6T2WH6R]], the [[space of convergent sequences|GWCBGYG5RA]] and the [[space of null sequences|WDH6QSH1Z6]].
Sequence spaces give the simplest examples of infinite-dimensional normed and topological vector spaces.
In summability theory, the sequences summable by a matrix method form a sequence space, and functional analysis is used to study it.

## Explanations

### L1 human

Think of a sequence as a vector with infinitely many coordinates.
Adding two sequences term by term and multiplying a sequence by a number work just as for vectors in $\R^n$.
A sequence space is any collection of sequences that is closed under these two operations.
For example, the sequences that tend to $0$ form a sequence space, because the sum of two such sequences also tends to $0$.