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

The L p, 1 ≤ p ≤ ∞ L^p, 1 p, 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^ (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.