In mathematics, the H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus.
Let
i
:
H
→
E
be an abstract Wiener space, and suppose that
F
:
E
→
R
is differentiable. Then the Fréchet derivative is a map
D
F
:
E
→
L
i
n
(
E
;
R
)
;
i.e., for
x
∈
E
,
D
F
(
x
)
is an element of
E
∗
, the dual space to
E
.
Therefore, define the
H
-derivative
D
H
F
at
x
∈
E
by
D
H
F
(
x
)
:=
D
F
(
x
)
∘
i
:
H
→
R
,
a continuous linear map on
H
.
Define the
H
-gradient
∇
H
F
:
E
→
H
by
⟨
∇
H
F
(
x
)
,
h
⟩
H
=
(
D
H
F
)
(
x
)
(
h
)
=
lim
t
→
0
F
(
x
+
t
i
(
h
)
)
−
F
(
x
)
t
.
That is, if
j
:
E
∗
→
H
denotes the adjoint of
i
:
H
→
E
, we have
∇
H
F
(
x
)
:=
j
(
D
F
(
x
)
)
.
