aarjaneiro/planetmath huggingface dataset:
Sobolev space

Sobolev space

We define the

Sobolev spaces

of functions

W

m

,

p

⁢

(

Ω

)

where

Ω

is an

open subset

of

𝐑

n

,

m

≥

0

is an integer and

p

∈

[

1

,

+

∞

]

.

The spaces

W

0

,

p

⁢

(

Ω

)

are simply defined to be the spaces

L

p

⁢

(

Ω

)

of Lebesgue

p

-summable functions.
We then define the space

W

m

,

p

⁢

(

Ω

)

to be the

space of functions

u

∈

L

p

⁢

(

Ω

)

which have

weak derivatives

g

=

(

g

1

,

…

,

g

n

)

such that

g

i

∈

W

m

-

1

,

p

⁢

(

Ω

)

.

The space

W

m

,

p

turns out to be a

Banach space

when endowed with the norm

∥

u

∥

W

m

,

p

=

∑

k

=

0

m

∑

i

1

=

1

n

⋯

⁢

∑

i

k

=

1

n

[

∫

Ω

|

∂

k

⁡

u

⁢

(

x

)

∂

⁡

x

i

1

⁢

⋯

⁢

∂

⁡

x

i

k

|

p

⁢

𝑑

x

]

1

p

i.e. the sum of the

L

p

norms of

u

and of all weak derivatives of

u

up to the

m

-th order.

Of particular interest are the spaces

H

m

⁢

(

Ω

)

:=

W

m

,

2

⁢

(

Ω

)

which turn out to be

Hilbert spaces

with the

scalar product

given by

(

u

,

v

)

H

m

⁢

(

Ω

)

=

∑

k

=

0

m

∑

i

1

=

1

n

⋯

⁢

∑

i

k

=

1

n

∫

Ω

∂

k

⁡

u

⁢

(

x

)

∂

⁡

x

i

1

⁢

⋯

⁢

∂

⁡

x

i

k

⁢

∂

k

⁡

v

⁢

(

x

)

∂

⁡

x

i

1

⁢

⋯

⁢

∂

⁡

x

i

k

⁢

𝑑

x

.

Title

Sobolev space

Canonical name

SobolevSpace

Date of creation

2013-03-22 14:54:55

Last modified on

2013-03-22 14:54:55

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

10

Author

paolini (1187)

Entry type

Definition

Classification

msc 46E35

Synonym

Sobolev function

Related topic

WeakDerivative

Generated on Fri Feb 9 21:40:19 2018 by

LaTeXML
weak derivative

weak derivative

Let

f

:

Ω

→

𝐑

and

g

=

(

g

1

,

…

,

g

n

)

:

Ω

→

𝐑

n

be

locally integrable
functions

defined on an open set

Ω

⊂

𝐑

n

.
We say that

g

is the

weak derivative

of

f

if the equality

∫

Ω

f

⁢

∂

⁡

ϕ

∂

⁡

x

i

=

-

∫

Ω

g

i

⁢

ϕ

holds true for all

functions

ϕ

∈

𝒞

c

∞

⁢

(

Ω

)

(

smooth functions with compact support

in

Ω

) and for all

i

=

1

,

…

,

n

. Notice that the

integrals

involved are well defined since

ϕ

is

bounded

with compact support and because

f

is assumed to be

integrable

on

compact subsets

of

Ω

.

Comments

1.

If

f

is of class

𝒞

1

then the

gradient

of

f

is the weak derivative of

f

in view of

Gauss Green Theorem

. So the weak derivative is an

extension

of the classical

derivative

.

2.

The weak derivative is unique (as an element of the Lebesgue space

L

l

⁢

o

⁢

c

1

) in view of a result about locally integrable functions.

3.

The same definition can be given for functions with complex values.

Title

weak derivative

Canonical name

WeakDerivative

Date of creation

2013-03-22 14:54:52

Last modified on

2013-03-22 14:54:52

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

16

Author

paolini (1187)

Entry type

Definition

Classification

msc 46E35

Related topic

SobolevSpaces

Generated on Fri Feb 9 21:40:21 2018 by

LaTeXML
Cauchy principal part integral

Cauchy principal part integral

Definition

[

1

,

2

,

3

]

Let

C

0

∞

⁢

(

ℝ

)

be the set of

smooth functions with compact support

on

ℝ

.
Then the

Cauchy principal part integral

(or, more in line with the notation, the

Cauchy principal value

)

p

.

v

.

⁡

(

1

x

)

is

mapping

p

.

v

.

⁡

(

1

x

)

:

C

0

∞

⁢

(

ℝ

)

→

ℂ

defined as

p

.

v

.

⁡

(

1

x

)

⁢

(

u

)

=

lim

ε

→

0

+

⁡

∫

|

x

|

>

ε

u

⁢

(

x

)

x

⁢

𝑑

x

for

u

∈

C

0

∞

⁢

(

ℝ

)

.

Theorem

The mapping

p

.

v

.

⁡

(

1

x

)

is a

distribution

of

first order

(

http://planetmath.org/Distribution4

).
That is,

p

.

v

.

⁡

(

1

x

)

∈

𝒟

′

⁣

1

⁢

(

ℝ

)

.

(proof. (

http://planetmath.org/Operatornamepvfrac1xIsADistributionOfFirstOrder

))

0.0.1

Properties

1.

The distribution

p

.

v

.

⁡

(

1

x

)

is obtained as the limit (

[

3

]

, pp. 250)

χ

n

⁢

|

x

|

x

→

p

.

v

.

⁡

(

1

x

)

.

as

n

→

∞

.
Here,

χ

is the

characteristic function

,
the

locally integrable functions

on the left hand side
should be interpreted as distributions
(see this page (

http://planetmath.org/EveryLocallyIntegrableFunctionIsADistribution

)),
and the limit should be taken in

𝒟

′

⁢

(

ℝ

)

. It should also be noted that

p

.

v

.

⁡

(

1

x

)

can be represented by a proper integral as

p

.

v

.

⁡

(

1

x

)

⁢

(

u

)

=

∫

0

∞

u

⁢

(

x

)

-

u

⁢

(

-

x

)

x

,

where we have used the fact that the integrand is

continuous

because of the differentiability at 0. In fact, this viewpoint can be used to somewhat vastly increase the set of

functions

for which this

principal value

is

well-defined

, such as functions that are

integrable

,

satisfy

a

Lipschitz condition

at 0, and whose behavior for large

x

makes the

integral

converge

at

infinity

.

2.

Let

ln

⁡

|

t

|

be the distribution induced by the
locally integrable function

ln

⁡

|

t

|

:

ℝ

→

ℝ

. Then,
for the distributional

derivative

(

http://planetmath.org/OperationsOnDistributions

)

D

,
we have (

[

2

]

, pp. 149)

D

⁢

(

ln

⁡

|

t

|

)

=

p

.

v

.

⁡

(

1

x

)

.

References

1

M. Reed, B. Simon,

Methods of Modern Mathematical Physics:

Functional Analysis

I

,
Revised and enlarged edition, Academic Press, 1980.

2

S. Igari,

Real analysis

- With an

introduction

to

Wavelet

Theory

, American Mathematical Society, 1998.

3

J. Rauch,

Partial Differential Equations

, Springer-Verlag, 1991.

Title

Cauchy principal part integral

Canonical name

CauchyPrincipalPartIntegral

Date of creation

2013-03-22 13:46:04

Last modified on

2013-03-22 13:46:04

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Definition

Classification

msc 46F05

Classification

msc 46-00

Synonym

Cauchy principal value

Related topic

ImproperIntegral

Generated on Fri Feb 9 21:40:27 2018 by

LaTeXML
space of rapidly decreasing functions

space of rapidly decreasing functions

The

function space

of rapidly

decreasing functions

𝒮

has the
important

property

that the

Fourier transform

is an endomorphism
on this space. This property enables one, by duality, to
define the Fourier transform
for elements in the

dual space

of

𝒮

, that is, for

tempered
distributions

.

Definition

The

space of rapidly decreasing functions

on

ℝ

n

is the function space

𝒮

(

ℝ

n

)

=

{

f

∈

C

∞

(

ℝ

n

)

∣

sup

x

∈

ℝ

n

∣

|

|

f

|

|

α

,

β

<

∞

for all multi-indices

α

,

β

}

,

where

C

∞

⁢

(

ℝ

n

)

is the set of

smooth functions

from

ℝ

n

to

ℂ

, and

||

f

||

α

,

β

=

||

x

α

⁢

D

β

⁢

f

||

∞

.

Here,

|

|

⋅

|

|

∞

is the

supremum norm

, and we use
multi-index notation.
When the

dimension

n

is clear, it is convenient to write

𝒮

=

𝒮

⁢

(

ℝ

n

)

. The space

𝒮

is also called the

Schwartz space

, after Laurent Schwartz
(1915-2002)

[

2

]

.

0.0.1

Examples of functions in

𝒮

1.

If

i

is a multi-index, and

a

is a positive
real number, then

x

i

⁢

exp

⁡

{

-

a

⁢

x

2

}

∈

𝒮

.

2.

Any

smooth function with compact support

f

is in

𝒮

.
This is clear since any

derivative

of

f

is

continuous

, so

x

α

⁢

D

β

⁢

f

has a

maximum

in

ℝ

n

.

0.0.2

Properties

1.

𝒮

is a

complex vector space

. In other words,

𝒮

is closed under point-wise

addition

and under

multiplication

by a complex scalar.

2.

Using Leibniz’ rule, it follows that

𝒮

is also closed
under point-wise multiplication; if

f

,

g

∈

𝒮

, then

f

⁢

g

:

x

↦

f

⁢

(

x

)

⁢

g

⁢

(

x

)

is also in

𝒮

.

3.

For any

1

≤

p

≤

∞

, we have

[

3

]

𝒮

⊂

L

p

,

and if

p

<

∞

, then

𝒮

is also dense in

L

p

.

4.

The Fourier transform is a

linear isomorphism

𝒮

→

𝒮

.

References

1

L. Hörmander,

The

Analysis

of Linear Partial Differential Operators I,
(

Distribution

theory and

Fourier Analysis

)

, 2nd ed, Springer-Verlag, 1990.

2

The MacTutor History of Mathematics archive,

http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Schwartz.html

Laurent Schwartz

3

M. Reed, B. Simon,

Methods of Modern

Mathematical Physics

:

Functional Analysis

I

,
Revised and enlarged edition, Academic Press, 1980.

4

Wikipedia,

http://en.wikipedia.org/wiki/Tempered_distribution

Tempered distributions

Title

space of rapidly decreasing functions

Canonical

name

SpaceOfRapidlyDecreasingFunctions

Date of creation

2013-03-22 13:44:50

Last modified on

2013-03-22 13:44:50

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

8

Author

matte (1858)

Entry type

Definition

Classification

msc 46F05

Synonym

Schwartz space

Related topic

DiscreteTimeFourierTransformInRelationWithItsContinousTimeFourierTransfrom

Generated on Fri Feb 9 21:40:33 2018 by

LaTeXML
T_f is a distribution of zeroth order

T

f

is a distribution of zeroth order

To check that

T

f

is a

distribution

of zeroth order (

http://planetmath.org/Distribution4

),
we shall use
condition (3) on this page (

http://planetmath.org/Distribution4

).
First, it is clear that

T

f

is a linear mapping.
To see that

T

f

is

continuous

, suppose

K

is a

compact set

in

U

and

u

∈

𝒟

K

, i.e.,

u

is a

smooth function

with

support

in

K

.
We then have

|

T

f

⁢

(

u

)

|

=

|

∫

K

f

⁢

(

x

)

⁢

u

⁢

(

x

)

⁢

𝑑

x

|

≤

∫

K

|

f

⁢

(

x

)

|

⁢

|

u

⁢

(

x

)

|

⁢

𝑑

x

≤

∫

K

|

f

⁢

(

x

)

|

⁢

𝑑

x

⁢

||

u

||

∞

.

Since

f

is locally integrable, it follows that

C

=

∫

K

|

f

⁢

(

x

)

|

⁢

𝑑

x

is finite, so

|

T

f

⁢

(

u

)

|

≤

C

⁢

||

u

||

∞

.

Thus

f

is a distribution of zeroth order (

[

1

]

, pp. 381).

□

References

1

S. Lang,

Analysis

II

,
Addison-Wesley Publishing Company Inc., 1969.

Title

T

f

is a distribution of zeroth order

Canonical name

TfIsADistributionOfZerothOrder

Date of creation

2013-03-22 13:44:28

Last modified on

2013-03-22 13:44:28

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Proof

Classification

msc 46F05

Classification

msc 46-00

Generated on Fri Feb 9 21:40:35 2018 by

LaTeXML
p.ERROR(\tmspace)-.1667emv.⁡(1/x) is a distribution of first order

p

.

\tmspace

-

.1667

emv

.

⁡

(

1

x

)

is a distribution of first order

(Following

[

1

,

2

]

.)
Let

u

∈

𝒟

⁢

(

U

)

. Then

supp

⁡

u

⊂

[

-

k

,

k

]

for
some

k

>

0

. For any

ε

>

0

,

u

⁢

(

x

)

/

x

is

Lebesgue integrable

in

|

x

|

∈

[

ε

,

k

]

. Thus, by a change of variable, we have

p

.

v

.

⁡

(

1

x

)

⁢

(

u

)

=

lim

ε

→

0

+

⁡

∫

[

ε

,

k

]

u

⁢

(

x

)

-

u

⁢

(

-

x

)

x

⁢

𝑑

x

.

Now it is clear that the

integrand

is

continuous

for all

x

∈

ℝ

∖

{

0

}

.
What is more, the integrand approaches

2

⁢

u

′

⁢

(

0

)

for

x

→

0

, so the
integrand has a

removable discontinuity

at

x

=

0

. That is, by assigning the value

2

⁢

u

′

⁢

(

0

)

to the integrand at

x

=

0

, the integrand becomes continuous in

[

0

,

k

]

.
This means that the integrand is

Lebesgue measurable

on

[

0

,

k

]

.
Then, by defining

f

n

⁢

(

x

)

=

χ

[

1

/

n

,

k

]

⁢

(

u

⁢

(

x

)

-

u

⁢

(

-

x

)

)

/

x

(where

χ

is the

characteristic function

), and
applying the

Lebesgue dominated convergence theorem

, we have

p

.

v

.

⁡

(

1

x

)

⁢

(

u

)

=

∫

[

0

,

k

]

u

⁢

(

x

)

-

u

⁢

(

-

x

)

x

⁢

𝑑

x

.

It follows that

p

.

v

.

⁡

(

1

x

)

⁢

(

u

)

is finite, i.e.,

p

.

v

.

⁡

(

1

x

)

takes values in

ℂ

.
Since

𝒟

⁢

(

U

)

is a vector space, if follows easily from the above

expression

that

p

.

v

.

⁡

(

1

x

)

is linear.

To prove that

p

.

v

.

⁡

(

1

x

)

is continuous, we shall use condition (3) on
this page (

http://planetmath.org/Distribution4

).
For this, suppose

K

is a

compact subset

of

ℝ

and

u

∈

𝒟

K

. Again, we can assume that

K

⊂

[

-

k

,

k

]

for some

k

>

0

.
For

x

>

0

, we have

|

u

⁢

(

x

)

-

u

⁢

(

-

x

)

x

|

=

|

1

x

⁢

∫

(

-

x

,

x

)

u

′

⁢

(

t

)

⁢

𝑑

t

|

≤

2

⁢

||

u

′

||

∞

,

where

|

|

⋅

|

|

∞

is the

supremum norm

. In the first equality we have used the

fundamental theorem of calculus

for the

Lebesgue integral

(valid since

u

is

absolutely continuous

on

[

-

k

,

k

]

). Thus

|

p

.

v

.

⁡

(

1

x

)

⁢

(

u

)

|

≤

2

⁢

k

⁢

||

u

′

||

∞

and

p

.

v

.

⁡

(

1

x

)

is a

distribution

of

first order

(

http://planetmath.org/Distribution4

) as claimed.

□

References

1

M. Reed, B. Simon,

Methods of Modern

Mathematical Physics

:

Functional Analysis

I

,
Revised and enlarged edition, Academic Press, 1980.

2

S. Igari,

Real analysis

- With an

introduction

to

Wavelet

Theory

, American Mathematical Society, 1998.

Title

p

.

\tmspace

-

.1667

emv

.

⁡

(

1

x

)

is a distribution of first order

Canonical

name

operatornamepvfrac1xIsADistributionOfFirstOrder

Date of creation

2013-03-22 13:46:07

Last modified on

2013-03-22 13:46:07

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Proof

Classification

msc 46F05

Classification

msc 46-00

Generated on Fri Feb 9 21:40:38 2018 by

LaTeXML
C^n

C

n

Let

f

:

ℝ

→

ℝ

be a

function

. We say that

f

is of class

C

1

if

f

′

exists and is

continuous

.

We also say that

f

is of class

C

n

if its

n

-th

derivative

exists and is continuous (and therefore all other previous derivatives exist and are continuous too).

The class of

continuous functions

is denoted by

C

0

. So we get the
following relationship among these classes:

C

0

⊃

C

1

⊃

C

2

⊃

C

3

⊃

…

Finally, the class of functions that have continuous derivatives of any order is denoted by

C

∞

and thus

C

∞

=

⋂

n

=

0

∞

C

n

.

It holds that any function that is

differentiable

is also continuous
(see this entry (

http://

planetmath

.org/DifferentiableFunctionsAreContinuous

)).
Therefore,

f

∈

C

∞

if and only if every derivative of

f

exists.

The previous

concepts

can be extended to functions

f

:

ℝ

m

→

ℝ

,
where

f

being of class

C

n

amounts to asking that all the

partial derivatives

of order

n

be continuous.
For

instance

,

f

:

ℝ

m

→

ℝ

being

C

2

means that

∂

2

⁡

f

∂

⁡

x

j

⁢

∂

⁡

x

i

exists and are all continuous for any

i

,

j

from

1

to

m

.

C

n

functions on an open set of

ℝ

m

Sometimes we need to talk about

continuity

not globally on

ℝ

,
but on some

interval

or open set.

If

U

⊆

ℝ

m

is an open set, and

f

:

U

→

ℝ

(or

f

:

U

→

ℂ

)
we say that

f

is of class

C

n

if

∂

α

⁡

f

exist and are continuous for all multi-indices

α

with

|

α

|

≤

n

.
See this page (

http://planetmath.org/MultiIndexNotation

) for the multi-index notation.

Title

C

n

Canonical name

Cn

Date of creation

2013-03-22 14:59:43

Last modified on

2013-03-22 14:59:43

Owner

drini (3)

Last modified by

drini (3)

Numerical id

13

Author

drini (3)

Entry type

Definition

Classification

msc 46G05

Classification

msc 26B05

Classification

msc 26A99

Classification

msc 26A24

Classification

msc 26A15

Synonym

C

1

Synonym

C

2

Synonym

C

k

Synonym

C

∞

Related topic

Derivative

Related topic

SmoothFunctionsWithCompactSupport

Generated on Fri Feb 9 21:40:44 2018 by

LaTeXML
C^n norm

C

n

norm

One can define an

extended norm

on the space

C

n

⁢

(

I

)

where

I

is a subset of

ℝ

as follows:

∥

f

∥

C

n

=

sup

x

∈

I

⁡

sup

k

≤

n

⁡

|

d

k

⁢

f

d

⁢

x

k

|

If

f

is a

function

of more than one variable (i.e. lies in

C

n

⁢

(

D

)

for a subset

D

∈

ℝ

m

), then one needs to take the

supremum

over all

partial derivatives

of order up to

n

.

That

∥

⋅

∥

C

n

satisfies

the defining conditions for an extended norm follows trivially from the properties of the

absolute value

(positivity, homogeneity, and the

triangle inequality

) and the

inequality

sup

⁡

(

|

f

|

+

|

g

|

)

<

sup

⁡

|

f

|

+

sup

⁡

|

g

|

.

If we are considering functions defined on the whole of

ℝ

m

or an

unbounded

subset of

ℝ

m

, the

C

n

norm may be

infinite

. For example,

∥

e

x

∥

C

n

=

∞

for all

n

because the

n

-th

derivative

of

e

x

is again

e

x

, which blows up as

x

approaches

infinity

. If we are considering functions on a

compact

(closed and

bounded

) subset of

ℝ

m

however, the

C

n

norm is always finite as a

consequence

of the fact that every

continuous function

on a compact set attains a

maximum

. This also means that we may replace the “

sup

” with a “

max

” in our definition in this case.

Having a

sequence

of functions

converge

under this norm is the same as having their

n

-th derivatives converge uniformly. Therefore, it follows from the fact that the uniform limit of continuous functions is

continuous

that

C

n

is

complete

under this norm. (In other words, it is a

Banach space

.)

In the case of

C

∞

, there is no natural way to impose a norm, so instead one uses all the

C

n

norms to define the

topology

in

C

∞

. One does this by declaring that a subset of

C

∞

is closed if it is closed in all the

C

n

norms. A space like this whose topology is defined by an infinite

collection

of norms is known as a multi-normed space.

Title

C

n

norm

Canonical name

CnNorm

Date of creation

2013-03-22 14:59:46

Last modified on

2013-03-22 14:59:46

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 46G05

Classification

msc 26B05

Classification

msc 26Axx

Classification

msc 26A24

Classification

msc 26A15

Generated on Fri Feb 9 21:40:46 2018 by

LaTeXML
criteria for existence of antidervatives

criteria for existence of antidervatives

Let

X

be a

normed space

,

Y

a

Banach space

,

U

⊂

X

a

connected

open set

,

f

:

U

→

L

⁢

(

X

;

Y

)

a

continuous function

, where

L

⁢

(

X

;

Y

)

is the space of continuous linear operators. In this article a path is a curve that has

bounded variation

. The following

theorems

give necessary and

sufficient

conditions for

f

to have an

antiderivatives

.

Theorem 1

.

The following conditions are

equivalent

:

1.

f

has an antiderivative on

U

,

2.

for any

γ

closed path in

U

∫

γ

f

=

0

,

3.

for any

γ

,

δ

paths in

U

that have the same starting and

endpoints

∫

γ

f

=

∫

δ

f

.

The next theorem states criteria for the

existence

of local antiderivatives.

Theorem 2

.

The following conditions are equivalent:

1.

f

has an antiderivative locally,

2.

for

γ

,

δ

homotopic closed paths in

U

∫

γ

f

=

∫

δ

f

,

3.

if

γ

is a triangular path such that its convex hull is in

U

, then

∫

γ

f

=

0

.

With the

stronger

assumption

that

f

is differenciable we can obtain a more easily applicable condition. We introduce the canonical

isometric isomorphism

π

1

,

1

:

L

(

X

;

L

(

X

;

Y

)

)

→

L

2

(

X

;

Y

)

,

u

↦

(

(

x

1

,

x

2

)

↦

u

(

x

1

)

(

x

2

)

)

where

L

2

⁢

(

X

;

Y

)

is the space of bilinear operators from

X

to

Y

. If

F

is an antiderivative of

f

, then

π

1

,

1

⁢

(

D

⁢

f

⁢

(

x

)

)

=

D

2

⁢

F

⁢

(

x

)

and by Clairaut’s theorem the

second derivative

is

symmetric

. The following theorems assert that the reverse is also true.

Theorem 3

.

If

f

is

differentiable

, then it has an antiderivative locally if and only if

π

1

,

1

⁢

(

D

⁢

f

⁢

(

x

)

)

is symmetric for all

x

∈

U

.

Combining these three theorems immediately gives the following.

Corollary 1

.

If

U

is simply connected and

f

is differentiable, then it has an antiderivative on

U

if and only if

π

1

,

1

⁢

(

D

⁢

f

⁢

(

x

)

)

is symmetric for all

x

∈

U

.

Title

criteria for existence of antidervatives

Canonical name

CriteriaForExistenceOfAntidervatives

Date of creation

2013-03-22 19:14:00

Last modified on

2013-03-22 19:14:00

Owner

scineram (4030)

Last modified by

scineram (4030)

Numerical id

9

Author

scineram (4030)

Entry type

Theorem

Classification

msc 46G05

Generated on Fri Feb 9 21:40:48 2018 by

LaTeXML
Fréchet derivative is unique

Fréchet derivative is unique

Theorem

The Fréchet

derivative

is unique.

Proof.

Assume that both

A

and

B

in

L

⁢

(

𝖵

,

𝖶

)

satisfy the condition for the Fréchet derivative (

http://planetmath.org/derivative2

) at the point

𝐱

. To prove that they are equal we will show that for all

ε

>

0

the

operator norm

∥

A

-

B

∥

is not greater than

ε

. By the

definition

of limit there exists a

positive

δ

such that for all

∥

𝐡

∥

≤

δ

∥

f

⁢

(

𝐱

+

𝐡

)

-

f

⁢

(

𝐱

)

-

A

⁢

𝐡

∥

≤

ε

2

⋅

∥

𝐡

∥

⁢

and

⁢

∥

f

⁢

(

𝐱

+

𝐡

)

-

f

⁢

(

𝐱

)

-

B

⁢

𝐡

∥

≤

ε

2

⋅

∥

𝐡

∥

holds. This gives

∥

(

A

-

B

)

⁢

𝐡

∥

=

∥

(

f

⁢

(

𝐱

+

𝐡

)

-

f

⁢

(

𝐱

)

-

A

⁢

𝐡

)

-

(

f

⁢

(

𝐱

+

𝐡

)

-

f

⁢

(

𝐱

)

-

B

⁢

𝐡

)

∥

≤

∥

f

⁢

(

𝐱

+

𝐡

)

-

f

⁢

(

𝐱

)

-

A

⁢

𝐡

∥

+

∥

f

⁢

(

𝐱

+

𝐡

)

-

f

⁢

(

𝐱

)

-

B

⁢

𝐡

∥

<

ε

⋅

∥

𝐡

∥

.

Now we have

δ

⋅

∥

A

-

B

∥

=

δ

⋅

sup

∥

𝐠

∥

≤

1

⁡

∥

(

A

-

B

)

⁢

𝐠

∥

=

sup

∥

𝐠

∥

≤

δ

⁡

∥

(

A

-

B

)

⁢

𝐠

∥

≤

sup

∥

𝐠

∥

≤

δ

⁡

ε

⋅

∥

𝐠

∥

≤

ε

⋅

δ

,

thus

∥

A

-

B

∥

≤

ε

as we wanted to show.

Title

Fréchet derivative is unique

Canonical

name

FrechetDerivativeIsUnique

Date of creation

2013-03-22 16:08:35

Last modified on

2013-03-22 16:08:35

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

12

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 46G05

Related topic

derivative

Generated on Fri Feb 9 21:40:51 2018 by

LaTeXML