---
tag: 9240K052CS
kind: concept
name: Quotient of a normed space
msc: [46B99]
specializes: [0BR3FY5Q8M]
sources:
  - url: https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra)
---

## Summary

The quotient of a normed space $X$ by a closed subspace $M$ is the [[quotient vector space|0BR3FY5Q8M]] $X/M$.
Its norm measures the distance of a vector to $M$.
It is a normed space, and it is a [[Banach space|06GZ956657]] when $X$ is.
Quotients appear whenever one ignores a subspace.
Examples are the open mapping theorem, the study of the kernel of an operator, and duality, where $(X/M)^*$ is identified with the annihilator of $M$.

## Explanations

### L1 human

To form the quotient, one declares two vectors equal when their difference lies in $M$.
The size of a class is the shortest length of a vector in that class, that is, the distance from the class to the origin.
In the plane with the usual length, take $M$ to be the horizontal axis.
Each class is a horizontal line, and its size is its height above or below the axis.