Riesz space

Remarks

Example: $\R^n$ with the componentwise order is a Riesz space, with $x \vee y$ the componentwise maximum. Example: the space $C([0, 1])$ of continuous functions with the pointwise order is a Riesz space, since $\max(f, g)$ and $\min(f, g)$ are continuous. Non-example: on $\R^2$, let $(a, b) \le (c, d)$ when $a \le c$ and $b = d$. This makes $\R^2$ an ordered vector space, but $(0, 0)$ and $(0, 1)$ have no common upper bound, so it is not a lattice. In a Riesz space one writes $x^+ = x \vee 0$, $x^- = (-x) \vee 0$ and $\abs{x} = x \vee (-x)$, and then $x = x^+ - x^-$ and $\abs{x} = x^+ + x^-$. A Riesz space with a complete norm satisfying $\abs{x} \le \abs{y} \implies \norm{x} \le \norm{y}$ is a Banach lattice.