Approximation Theorems in Locally Convex Lattices

Abstract

We recall that an order relation “≤” on a real vector E is called compatible with the linear structure of the space if: $$\displaystyle (\mathrm{i})_{ }^{} { }_{ }^{} \forall \; \; x,y\in E,\; \; x\le y\; \Rightarrow \; x+z\le y+z,_{ }^{} \forall \; z\in E, $$ $$\displaystyle (\mathrm{ii})_{ }^{} \; \forall \; x,y\in E,_{ }^{} \; x\le y,_{ }^{} \Rightarrow \; \; \alpha \cdot x\le \alpha \cdot y \; \; ,\forall \alpha \in \mathrm {R}_{+}. $$ An ordered vector space is a real vector space endowed with an order relation compatible with the linear structure of the space. An ordered set E is called lattice if for any x, y ∈ E there exists the upper bound x ∨ y ∈ E and the lower bound x ∧ y ∈ E. A vector lattice is any ordered vector space that is a lattice. Let E be a vector lattice.