Notes on Some Convergences in Riesz Space

An equivalent description of u-uniform convergence is presented first. Then the relations among the order convergence, u-uniform convergence and norm convergence of sequences are discussed in Riesz spaces. An equivalence of the three convergences is brought forward; namely, {fn} is a u-uniform Cauchy sequence. Finally the relations among the three convergences of sequences are also extended to the relations among the convergences of nets in Riesz spaces.