The Banach space
L
1
(
0
,
T
;
X
)
{L^1}(0,T;X)
is retopologized by
|
|
|
f
|
|
|
=
max
|
|
∫
a
b
f
d
t
|
|
|||f||| = \max ||\int _a^bfdt||
,
0
⩽
a
⩽
b
⩽
T
0 \leqslant a \leqslant b \leqslant T
, where
|
|
.
|
|
||.||
is the norm in the given Banach space
X
X
. It is shown here that this topology coincides with the usual weak topology of
L
1
(
0
,
T
;
X
)
{L^1}(0,T;X)
on a wide class of weakly compact subsets.