Abstract :
In this paper a simple proof for the following theorem, due to Luxemburg and Zaanen is given: an
Archimedean vector lattice A is Dedekind σ-complete if and only if A has the principal projection property
and A is uniformly complete. As an application, we give a new and short proof for the following
version of Freudenthal’s spectral theorem: let A be a uniformly complete vector lattice with the principal
projection property and let 0 < u ∈ A. For any element w in A such that 0 w u there exists a
sequence {sn: n = 1, 2, . . .} in A which satisfies 0 sn w (u), where each element sn is of the form
k
i=1 αipi , with real numbers α1, . . . , αk such that 0 αi 1 (i = 1, . . . , k) and mutually disjoint components
p1, . . . , pk of u.
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