Riesz space

From Wikipedia, the free encyclopedia

In mathematics a Riesz space, lattice-ordered vector space or vector lattice is an ordered vector space where the order structure is a lattice.

Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper Sur la décomposition des opérations fonctionelles linéaires.

Riesz spaces are important in measure theory.

[edit] Examples

• The space of continuous real valued function on a set X with compact support with the partial order f ≤ g is a Riesz space.

[edit] Properties

• Riesz spaces are lattice ordered groups
• Every Riesz space is a distributive lattice

[edit] References

• Bourbaki, Nicolas; Elements of Mathematics: Integration. Chapters 1–6; ISBN 3-540-41129-1
• Riesz, Frigyes; Sur la décomposition des opérations fonctionelles linéaires , Atti congress. internaz. mathematici (Bologna, 1928) , 3 , Zanichelli (1930) pp. 143–148
• Sobolev, V. I. (2001), “Riesz space”, Encyclopædia of Mathematics, Springer, ISBN 978-1-4020-0609-8, <http://eom.springer.de/R/r082290.htm>