Let
\,E\,
E
be an ideal of
\,L^0\,
L
0
over a
\,\sigma
σ
-finite measure space
\,(\Omega,\Sigma,\mu)\,
(
Ω
,
Σ
,
μ
)
and let
E'
E
′
be the Köthe dual of
\,E
E
. Let
\,(X,\|\cdot\|_X)\,
(
X
,
∥
⋅
∥
X
)
be a real Banach space, and
\,X^*\,
X
∗
the Banach dual of
\,X
X
. Let
\,E(X)\,
E
(
X
)
be a subspace of the space
\,L^0(X)\,
L
0
(
X
)
of
\mu
μ
-equivalence classes of all strongly
\Sigma
Σ
-measurable function
\def\si{\sigma}\def\Si{\Sigma}\def\om{\omega}\def\Om{\Omega}\def\ps{\rightarrow}\def\wf{\widetilde{f}}\def\wg{\widetilde{g}}\def\cl{{\cal L}}\,f:\Om\ps X
f
:
Ω
→
X
, and consisting of all those
\,f\in L^0(X)\,
f
∈
L
0
(
X
)
for which the scalar function
\widetilde{f}
f
~
, defined by
\,\widetilde{f}(\omega)=\|f(\omega)\|_X\,
f
~
(
ω
)
=
∥
f
(
ω
)
∥
X
for
\,\omega\in{\Omega}
ω
∈
Ω
, belongs to
E
E
. Assume that a Banach space
\,X\,
X
is an Asplund space. It is shown that a subset
C
C
of
\,E'(X^*)\,
E
′
(
X
∗
)
is relatively
\,\sigma(E'(X^*),E(X))
σ
(
E
′
(
X
∗
)
,
E
(
X
)
)
-compact iff the set
\,\{\widetilde{g}:g\in E'(X^*)\}\,
{
g
~
:
g
∈
E
′
(
X
∗
)
}
in
E'
E
′
is relatively
\,\sigma(E',E)
σ
(
E
′
,
E
)
-compact. We consider the topology
\,\overline{\tau(E,E')}\,
τ
(
E
,
E
′
)
‾
on
E(X)
E
(
X
)
associated with the Mackey topology
\,\tau(E,E')\,
τ
(
E
,
E
′
)
on
E
E
. It is shown that
\,\overline{\tau(E,E')}\,
τ
(
E
,
E
′
)
‾
is strongly Mackey topology; hence
\,\overline{\tau(E,E')}\,
τ
(
E
,
E
′
)
‾
coincides with the Mackey topology
\,\tau(E(X),E'(X^*))
τ
(
E
(
X
)
,
E
′
(
X
∗
)
)
. Moreover,
\,E'(X^*)\,
E
′
(
X
∗
)
is
\,\sigma(E'(X^*), E(X))
σ
(
E
′
(
X
∗
)
,
E
(
X
)
)
-sequentially complete whenever
E'
E
′
is perfect. We examine the space
{\cal L}_\tau(E(X),Y)
L
τ
(
E
(
X
)
,
Y
)
of all
\,(\tau(E(X),E'(X^*)),\|\cdot\|_Y)
(
τ
(
E
(
X
)
,
E
′
(
X
∗
)
)
,
∥
⋅
∥
Y
)
-continuous linear operators from
\,E(X)\,
E
(
X
)
to a Banach space
\,(Y,\|\cdot\|_Y)
(
Y
,
∥
⋅
∥
Y
)
, equipped with the weak operator topology (briefly WOT) and the strong operator topology (briefly SOT). It is shown that if
E
E
is perfect, then
{\cal L}_\tau(E(X),Y)
L
τ
(
E
(
X
)
,
Y
)
is WOT-sequentially complete, and every SOT-compact subset of
{\cal L}_\tau(E(X),Y)
L
τ
(
E
(
X
)
,
Y
)
is
\,(\tau(E(X),E'(X^*)),\|\cdot\|_Y)
(
τ
(
E
(
X
)
,
E
′
(
X
∗
)
)
,
∥
⋅
∥
Y
)
-equicontinuous. Moreover, a Vitali-Hahn-Saks type theorem for
{\cal L}_\tau(E(X),Y)
L
τ
(
E
(
X
)
,
Y
)
is obtained.