nLab
Riesz space

Definition

An ordered real vector space is a real vector space that has a compatible poset structure: if xyx\le y, then also x+ay+ax+a\le y+a for all aa and txtyt x\le t y for all real t0t\ge0.

A Riesz space is an ordered real vector space whose underlying poset is a lattice.

A morphism of Riesz spaces is a linear map that preserves finite infima and suprema.

A unital Riesz space is a Riesz space equipped with a unit: an element uu such that u0u\ge0 and for every xx there is an integer n0n\ge0 such that |x|=sup(x,x)nu|x|=\sup(x,-x) \le n u.

A morphism of unital Riesz spaces is a morphism of Riesz spaces that preserves the unit.

Any unit uu induces a seminorm: x u=inf{tt0and|x|tu}\|x\|_u=\inf\{t\mid t\ge0 \; and\; |x|\le t u\}.

An Archimedean Riesz space_ is a Riesz space such that any infinitesimal element ee is zero. Here ee is infinitesimal if there is bb such that nebn e\le b for all integer nn.

The seminorm induced by a unit on an Archimedean Riesz space is a norm.

An Archimedean unital Riesz space is uniformly complete if the norm induced by the unit (any unit, in fact) is complete.

References

The original definition is in

  • Frigyes Riesz, Sur la décomposition des opérations fonctionelles linéaires, Atti congress. internaz. mathematici (Bologna, 1928), 3, Zanichelli (1930), 143–148.

Last revised on June 15, 2021 at 08:19:23. See the history of this page for a list of all contributions to it.