---
tag: 8HEZ9TW1NK
kind: concept
name: BK-space
msc: [46B45, 46A45]
specializes: [71DMQHW6JZ]
sources:
  - url: https://en.wikipedia.org/wiki/BK-space
same_as:
  wikidata: Q4835784
  wikipedia: BK-space
---

## Summary

A BK-space is a [[sequence space|XYTN19EPF2]] with a norm that makes it a [[Banach space|06GZ956657]] in which each coordinate $x \mapsto x_n$ is continuous.
The letters stand for Banach and the German word Koordinate.
BK-spaces are exactly the [[FK-spaces|71DMQHW6JZ]] whose topology comes from a norm.
The spaces $\ell^p$, $c$ and $c_0$ with their usual norms are the basic examples.

## Explanations

### L1 human

A BK-space is a space of sequences with a way of measuring size, a norm, that behaves well.
The space is complete, so there are no "missing" limits.
And a small norm forces every single term of the sequence to be small.
For example, in the space of bounded sequences the norm is the largest absolute value of the terms, so each term is at most the norm.