In this section etc and are Banach spaces over is an open neighbourhood of .
An operator
is
called
-linear
and bounded if it is linear in each of its arguments and there exists a
such
that
The
norm of
is defined
by
Then
Every -linear bounded operator is continuous.
In the sequel we will write for
Suppose
The differential of
at
exists if there exists a bounded linear operator
such that
with
The second differential of
at
exists if there exists a bounded bilinear operator
such
that
with
Inductively we define
The
th
differential of
at
exists if there exists a bounded
-linear
operator
such
that
with
The differentials
are also dentoed by
We also write
=
and so on. The diferentials
are called the Frechet derivatives of
at
To formally find the Frechet derivatives we use the
formula
Once
we have a formula for the derivative we proceed to prove that it is the
desired one. This involves establishing two things:
the resulting operator is multilinear,bounded, and
the remainder is
is continuous at
if for each
there exists a
such that
This means that for all
and all
is called a function if is continuous on
If
has continuous paritial derivatives of order
then
is
and
where
() write Then
for some
Hence,
where
so
that
and
the continuity of the partial derivatives give that
(Bilinear Operators)
Suppose is a bilinear bounded operator. Then is
Set
Let's illustrate first the formal calculation of
and
Next,
we verify that these are the required derivatives. We begin with
Then i.e., is a bounded linear operator on into .
where Hence,
Similarly, for we have:
Then i.e., is a bounded bilinear operator on into .
where
Next we establish the continuity of
Hence,
i.e.,
is actually Lipschitz continuous. The continuity of
is easier to see since it is independent of
Finally,
Suppose
have Frechet derivatives of order
Then
Suppose
Let
and define the function
by
If
exists, we define the partial Frechet derivative
of
at
by
The
partial derivative
of
at
is defined in a similar manner. Partial derivatives are also denoted by
If
exists then
and
both exist and
With the above definition of
we
have
where
i.e.,
exists and
Hence,
Similarly,
Using
the linearity of
we obtain the assertion of the proposition.
Let
be a
-linear
bounded operator which is symmetric with respect ot all its arguments. This
means that
is invariant under any permutation of its argument. A power operator generated
by
is defined
by
and
where
(e.g., for since is symmetric.)
when
To see this, we write
has Lipschitz continuous derivatives of all orders
and
For
example, when
where
with
Also,
which shows the continuity of the operator
Hence,
The Lipschitz continuity of
can be shown as
follows:
for
Suppose
is a power operator for
Furthermore, suppose that the
series
for
all
The operator
defined by
is
called a power series. Note that () gurantees that
is a well defined element of
An operator is called analytic at the point if there exists a such that admits a power series representation () that is absolutely convergent (i.e., () converges) for every
If is analytic at the point then on for any where is as in Definition . Moreover, can be found by term by term differentiation of ().
We will show that
For
this we
have
and
where
Now,
for
Furthermore,
where
Since
as
then
Hence,
Therefore,
where
It remains to show that
This follows
from
The
statement about higher derivatives can be shown in the same manner.
Suppose
is a bounded bilinear operator,
such that
exist at some point
Then
is differentiable
and
where
Hence,
Denote the subset of
of operators with bounded inverse by
Define
by
Then
is analytic at
,
and
for
all
and all
For
let
.
Then
Since
the
series
converges, it follows that the series for
is absolutely convergent. Hence
is analytic at
To find
we use the formal
calculation
Now
noting that
is a bilinear operator, we use formula to compute higher derivatives of
.
For the second derivative we
get
This
formula can, in turn, be used to obtain higher derivatives of