Untitled

Abstract

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