Strong completeness of a class of 𝐿²(𝑇)-type Riesz spaces

Abstract

The L p , 1 ≤ p ≤ ∞ L^{p}, 1\le p\le \infty , spaces have been generalized to the setting of Riesz spaces as L p ( T ) {L}^{p}(T) spaces, on which there are R ( T ) R(T) -valued norms. The strong sequential completeness of the space L 1 ( T ) {L}^{1}(T) and the strong completeness of L ∞ ( T ) {L}^{\infty }(T) with resepct to their respective R ( T ) R(T) -valued norms were established by Kuo, Rodda, and Watson. In the current work, the T T -strong completeness of L 2 ( T ) {L}^{2}(T) is established via the Riesz–Fischer type theorem given by Kalauch, Kuo, and Watson. It is also shown that the conditional expectation operator T T is a weak order unit for the T T -strong dual.