Let
\,E\,
E
be an ideal of
\,L^{\rm o}\,
L
o
over a
\,\sigma
σ
-finite measure space
\,(\Omega, \Sigma, \mu)
(
Ω
,
Σ
,
μ
)
, and let
\,(X, \Vert\cdot {\Vert}_X)\,
(
X
,
∥
⋅
∥
X
)
be a real Banach space. Let
\,E(X)\,
E
(
X
)
be a subspace of the space
\,L^{\rm o}(X)\,
L
o
(
X
)
of
\,\mu
μ
-equivalence classes of all strongly
\,\Sigma
Σ
-measurable functions
\,f:\; \Omega\longrightarrow X\,
f
:
Ω
⟶
X
and consisting of all those
\,f\in L^{\rm o}(X)\,
f
∈
L
o
(
X
)
for which the scalar function
\,\Vert f(\cdot) \Vert_X\,
∥
f
(
⋅
)
∥
X
belongs to
\,E
E
. Let
\,E(X)_n^{\sim}\,
E
(
X
)
n
∼
stand for the order continuous dual of
\,E(X)
E
(
X
)
. In this paper we characterize both conditionally
\,\sigma(E(X),I)
σ
(
E
(
X
)
,
I
)
-compact and relatively
\,\sigma(E(X), I)
σ
(
E
(
X
)
,
I
)
-sequentially compact subsets of
\,E(X)\,
E
(
X
)
whenever
\,I\,
I
is an ideal of
\,E(X)_n^{\sim}
E
(
X
)
n
∼
. As an application, we obtain a characterization of almost reflexivity and reflexivity of a Banach space
\,X\,
X
in terms of conditionally
\,\sigma(E(X), I)
σ
(
E
(
X
)
,
I
)
-compact and relatively
\,\sigma(E(X), I)
σ
(
E
(
X
)
,
I
)
-sequentially compact subsets of
\,E(X)
E
(
X
)
.