  We give a Hahn-Jordan decomposition in Riesz spaces which generalizes that of
[{{\sc B. A. Watson}, {An And\^o-Douglas type theorem in Riesz spaces with a
conditional expectation,} {\em Positivity,} {\bf 13} (2009), 543 - 558}] and a
Riesz-Frechet representation theorem for the $T$-strong dual, where $T$ is a
Riesz space conditional expectation operator. The result of Watson was
formulated specifically to assist in the proof of the existence of Riesz space
conditional expectation operators with given range space, i.e., a result of
And\^{o}-Douglas type. This was needed in the study of Markov processes and
martingale theory in Riesz spaces. In the current work, our interest is a
Riesz-Frechet representation theorem, for which another variant of the
Hahn-Jordan decomposition is required.
