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.