# zbMATH — the first resource for mathematics

Completeness type properties of locally solid Riesz spaces. (English) Zbl 0543.46002
Let $$\Omega$$ be an extremally disconnected topological space, $$C^{\infty}(\Omega)$$ the Riesz space of continuous functions from $$\Omega$$ into $${\mathbb{R}}^{\infty}$$ which take finite values on dense subsets of $$\Omega$$, where $${\mathbb{R}}^{\infty}$$ denotes the real line compactified by $$+\infty$$ and -$$\infty$$. Function filters and topological vector cores are introduced and studied in $$C^{\infty}(\Omega)$$. These are in parallel with the function norm and Banach function spaces in the Luxemburg-Zaanen theory. Armed with this generalization, the following main results are proved.
Let $$(L,\tau)$$ be a Hausdorff topological Riesz space (the terminologies are as in C. D. Aliprantis and O. Burkinshaw’s book ’Locally Solid Riesz Spaces’ (1978; Zbl 0402.46005). $$(L,\tau)$$ embeds order densely into a Nakano space $$(L^{\#},\tau^{\#})$$ if and only if $$\tau$$ is Fatou; this embedding is unique up to an isomorphism. A Dedekind complete $$(L,\tau)$$ embeds order densely into a Hausdorff locally solid Dedekind complete Riesz space $$(L^{\#},\tau^{\#})$$ having the monotone completeness property if and only if $$\tau$$ is pseudo-Lebesgue.
Reviewer: Ng Kung-Fu

##### MSC:
 46A40 Ordered topological linear spaces, vector lattices
Full Text: