# Frechet Derivatives and the Implicit function theorem

In this section etc and are Banach spaces over is an open neighbourhood of .

### m-linear Bounded Operators

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

Proposition

Every -linear bounded operator is continuous.

In the sequel we will write for

### Frechet Derivatives

Definition

Suppose

1. The differential of at exists if there exists a bounded linear operator such that

with

2. The second differential of at exists if there exists a bounded bilinear operator such that

with

Inductively we define

3. 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:

1. the resulting operator is multilinear,bounded, and

2. the remainder is

Definition

is continuous at if for each there exists a such that

This means that for all and all

Definition

is called a function if is continuous on

Example

If has continuous paritial derivatives of order then is and

where

Proof

() write Then

for some Hence, where

so that

and the continuity of the partial derivatives give that

Example

(Bilinear Operators)

Suppose is a bilinear bounded operator. Then is

Proof

Set Let's illustrate first the formal calculation of and

Next, we verify that these are the required derivatives. We begin with

1. Then i.e., is a bounded linear operator on into .

2. where Hence,

Similarly, for we have:

1. Then i.e., is a bounded bilinear operator on into .

2. 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,

Proposition

Suppose have Frechet derivatives of order Then

### Partial Derivatives

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

Proposition

If exists then and both exist and

Proof

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.

## Application to Analytic Operators

### Power Operators

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

### Properties of Power Operators

1. (e.g., for since is symmetric.)

2. when

To see this, we write

3. 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

### Power Series

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

Definition

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

Theorem

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 ().

Proof

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.

#### A Useful Formula for Calculating Higher Derivatives

Proposition

Suppose is a bounded bilinear operator, such that exist at some point Then is differentiable and

Proof

where Hence,

Example

Denote the subset of of operators with bounded inverse by Define

by

Then is analytic at , and

for all and all

Proof

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

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