# nLab Riesz space

## Definition

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

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 $u$ such that $u\ge0$ and for every $x$ there is an integer $n\ge0$ such that $|x|=\sup(x,-x) \le n u$.

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

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

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

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.

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.