Two-Order Convergence - Belida, Joseph T (Archīum Ateneo, 2024)
A Riesz space or a vector lattice E is a partially ordered linear space that is also a lattice. Order convergence in a Riesz space will be defined and it will be shown that given a solid linear subspace X1 of a Riesz space X, a sequence {xn} is order convergent to x ϵ X1 if and only if the sequence {xn} is order convergent in X and order-bounded in X1. Here, a sequence is order-bounded in X1 if it has both upper and lower bounds in X1. With the convergence theorem above, familiar convergence theorems in analysis, such as the Lebesgue dominated convergence theorem, come in as easy examples. On the other hand, a two order convergence theorem will be presented and it will be shown that this will imply the known Controlled Convergence Theorem and that the Controlled-Convergence Theorem implies the Two-Order Convergence Theorem.