Taylor’s formula in Banach spaces

Taylor’s formula in Banach spaces

Let

U

be an open subset of a real

Banach space

X

.
If

f

:

U

→

ℝ

is

differentiable

n

+

1

times on

U

,
it may be

expanded

by Taylor’s formula:

f

⁢

(

x

)

=

f

⁢

(

a

)

+

D

⁡

f

⁢

(

a

)

⋅

h

+

1

2

!

⁢

D

2

⁡

f

⁢

(

a

)

⋅

h

2

+

⋯

+

1

n

!

⁢

D

n

⁡

f

⁢

(

a

)

⋅

h

n

+

R

n

⁢

(

x

)

,

(1)

with the following

expressions

for the remainder term

R

n

⁢

(

x

)

:

R

n

⁢

(

x

)

=

1

n

!

⁢

D

n

+

1

⁡

f

⁢

(

η

)

⋅

(

x

-

η

)

n

⁢

h

Cauchy form of

remainder

R

n

⁢

(

x

)

=

1

(

n

+

1

)

!

⁢

D

n

+

1

⁡

f

⁢

(

ξ

)

⋅

h

n

+

1

Lagrange form of remainder

R

n

⁢

(

x

)

=

1

n

!

⁢

∫

0

1

D

n

+

1

⁡

f

⁢

(

a

+

t

⁢

h

)

⋅

(

(

1

-

t

)

⁢

h

)

n

⁢

h

⁢

𝑑

t

integral form of remainder

Here

a

and

x

must be points of

U

such that the line segment
between

a

and

x

lie inside

U

,

h

is

x

-

a

,
and the points

ξ

and

η

lie on the same line segment,
strictly between

a

and

x

.

The

k

th Fréchet

derivative

of

f

at

a

is being denoted by

D

k

⁡

f

⁢

(

a

)

, to be viewed as a multilinear map

X

k

→

ℝ

.
The

⋅

h

k

notation means to evaluate a multilinear map
at

(

h

,

…

,

h

)

.

1

Remainders for vector-valued functions

If

Y

is a Banach space, we may also consider

Taylor expansions

for

f

:

U

→

Y

.
Formula (

4

) takes the same form,
but the Cauchy and Lagrange forms of the remainder
will not be exact;
they will only be bounds on

R

n

⁢

(

x

)

.
That is, for

f

:

U

→

Y

,

∥

R

n

⁢

(

x

)

∥

≤

1

n

!

⁢

∥

D

n

+

1

⁡

f

⁢

(

η

)

⋅

(

x

-

η

)

n

⁢

h

∥

Cauchy form of remainder

∥

R

n

⁢

(

x

)

∥

≤

1

(

n

+

1

)

!

⁢

∥

D

n

+

1

⁡

f

⁢

(

ξ

)

⋅

h

n

+

1

∥

Lagrange form of remainder

It is not hard to find

counterexamples

if we attempt to remove the norm signs or change
the

inequality

to equality in the above formulas.

However, the integral form of the remainder continues to hold for

Y

≠

ℝ

,
although

strictly

speaking it only applies if the

integrand

is

integrable

.
The integral form is also applicable when

X

and

Y

are

complex

Banach spaces.

Mean Value Theorem

The Mean Value Theorem can be obtained
as the special case

n

=

0

with the Lagrange form of the remainder:
for

f

:

U

→

Y

differentiable,

∥

f

⁢

(

x

)

-

f

⁢

(

a

)

∥

≤

∥

D

⁡

f

⁢

(

ξ

)

⋅

(

x

-

a

)

∥

(2)

If

Y

=

ℝ

, then the norm signs may be removed from
(

2

), and the inequality replaced by equality.

Formula (

2

) also holds under the much
weaker

hypothesis

that

f

only has a

directional derivative

along the line
segment from

a

to

x

.

Weaker bounds for the remainder

If

f

:

U

→

Y

is only differentiable

n

times

at

a

,
then we cannot quantify the remainder by the

n

+

1

th derivative,
but it is still true
that

R

n

⁢

(

x

)

=

o

⁢

(

∥

x

-

a

∥

n

)

⁢

as

x

→

a

.

(3)

Finite-dimensional case

If

X

=

ℝ

m

and

Y

=

ℝ

,

D

k

has the following expression in terms of coordinates:

D

k

⁡

f

⁢

(

a

)

⋅

(

ξ

1

,

…

,

ξ

k

)

=

∑

i

1

,

…

,

i

k

∂

k

⁡

f

∂

⁡

x

i

1

⁢

⋯

⁢

∂

⁡

x

i

k

⁢

ξ

1

i

1

⁢

⋯

⁢

ξ

k

i

k

,

where each

i

j

runs from

1

,

…

,

m

in the sum.

If we collect the equal mixed partials
(assuming that they are

continuous

)
then

1

k

!

⁢

D

k

⁡

f

⁢

(

a

)

⋅

h

k

=

∑

|

J

|

=

k

1

J

!

⁢

∂

|

J

|

⁡

f

∂

⁡

x

J

⁢

h

J

,

where

J

is a

multi-index

of

m

components

, and each component

J

i

indicates
how many times the derivative with respect to the

i

th coordinate should be taken,
and the

exponent

that the

i

th coordinate of

h

should be raised to
in the monomial

h

J

.
The multi-index

J

runs through all

combinations

such that

J

1

+

⋯

+

J

m

=

|

J

|

=

k

in the sum.
The notation

J

!

means

J

1

!

⁢

⋯

⁢

J

m

!

.

All this is more easily assimilated if we remember that

D

k

⁡

f

⁢

(

a

)

⋅

h

k

is supposed to be a polynomial of degree

k

.
Also

|

J

|

!

/

J

!

is just the

multinomial coefficient

.

Taylor series

If

lim

n

→

∞

⁡

R

n

⁢

(

x

)

=

0

,
then we may write

f

⁢

(

x

)

=

f

⁢

(

a

)

+

D

⁡

f

⁢

(

a

)

⋅

h

+

1

2

!

⁢

D

2

⁡

f

⁢

(

a

)

⋅

h

2

+

⋯

(4)

as a

convergent

infinite series. Elegant as such an expansion is,
it is not seen very often,
for the reason that higher order Fréchet derivatives, especially in infinite-dimensional spaces,
are often difficult to calculate.

But a notable exception occurs if a

function

f

is defined by a convergent “

power series

”

f

⁢

(

x

)

=

∑

k

=

0

∞

M

k

⋅

(

x

-

a

)

k

(5)

where

{

M

k

:

k

=

0

,

1

,

…

}

is a family of continuous symmetric multilinear functions

X

k

→

Y

.
In this case, the series (

5

) is the

Taylor series

for

f

at

a

.

References

1

Arthur Wouk.

A course of applied

functional analysis

. Wiley-Interscience, 1979.

2

Eberhard Zeidler.

Applied functional analysis: main principles and their applications

. Springer-Verlag, 1995.

3

Michael Spivak.

Calculus

, third edition. Publish or Perish, 1994.

Title

Taylor’s formula in Banach spaces

Canonical

name

TaylorsFormulaInBanachSpaces

Date of creation

2013-03-22 15:28:27

Last modified on

2013-03-22 15:28:27

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

10

Author

stevecheng (10074)

Entry type

Result

Classification

msc 46T20

Classification

msc 26B12

Classification

msc 41A58

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