---
tag: 71DMQHW6JZ
kind: concept
name: FK-space
msc: [46A45, 40H05]
specializes: [XYTN19EPF2]
sources:
  - url: https://en.wikipedia.org/wiki/FK-space
same_as:
  wikidata: Q5426006
  wikipedia: FK-space
---

## Summary

An FK-space is a [[sequence space|XYTN19EPF2]] with a complete metrizable locally convex topology in which each coordinate $x \mapsto x_n$ is continuous.
The letters stand for Fréchet and the German word Koordinate.
FK-spaces are the main tool of the functional analytic approach to summability: the domain of a matrix summability method is an FK-space.
A sequence space carries at most one FK topology, and inclusions between FK-spaces are automatically continuous.

## Explanations

### L1 human

A sequence space becomes much easier to study once it has a good notion of distance.
In an FK-space the distance is complete, like the distance on the real line, and it controls each term of a sequence.
So if two sequences are close in the space, then their first terms are close, their second terms are close, and so on.
The spaces $\ell^p$ with their usual norms are FK-spaces, and so is the space of all sequences with the topology of termwise convergence.