---
tag: 99709K2NJM
kind: concept
name: Riesz space
msc: ["46A40", "06F20"]
specializes: [HHMS3WH9T7, J0DHYQ0MDE]
sources:
  - url: https://en.wikipedia.org/wiki/Riesz_space
same_as:
  wikidata: Q2152570
  wikipedia: Riesz space
---

## Summary

A Riesz space, or vector lattice, is an [[ordered vector space|HHMS3WH9T7]] whose order is a [[lattice|WH4SFAACAS]].
It is at the same time a real vector space and a [[lattice-ordered group|J0DHYQ0MDE]].
Spaces of real functions with the pointwise order, such as spaces of continuous functions and $L^p$ spaces, are Riesz spaces.
Riesz spaces are the setting for positive operators and for [[Banach lattices|HDJ1GKG0NA]] in functional analysis.

## Explanations

### L1 human

In the space of real functions on an interval, you can add functions, scale them, and compare them point by point.
For two functions you can also take their pointwise maximum and minimum.
A Riesz space is a vector space with an order where these maxima and minima always exist and behave well with addition and scaling.
For instance, the absolute value $\abs{f} = \max(f, -f)$ of a function makes sense in any Riesz space.