Let
X
X
be a Banach space. Using derivatives in the sense of vector distributions, we show that the space
DC([0,1],X)
D
C
(
[
0
,
1
]
,
X
)
of all d.c. mappings from
[0,1]
[
0
,
1
]
into
X
X
, in a natural norm, is isomorphic to the space
M_{bv}([0,1], X)
M
b
v
(
[
0
,
1
]
,
X
)
of all vector measures with bounded variation. The same is proved for the space
BDC_b((0,\infty), X)
B
D
C
b
(
(
0
,
∞
)
,
X
)
of all bounded d.c. mappings with a bounded control function. The result for the space
DC([0,1], \mathbb{R})
D
C
(
[
0
,
1
]
,
R
)
of all continuous d.c. functions was (essentially) proved by M. Zippin [The space of differences of convex functions on
[0,1]
[
0
,
1
]
, Serdica Math. J. 26 (2000) 331–352] by a quite different method. The space
BDC_b((0,\infty), \mathbb{R})
B
D
C
b
(
(
0
,
∞
)
,
R
)
consists of all differences of two bounded convex functions. Internal characterizations of its members were given by O. Böhme [On functions which are the difference of two bounded convex functions on
(0,\infty)
(
0
,
∞
)
, Math. Nachr. 122 (1985) 45–58], but our characterization of its Banach structure is new.