Banach space valued analytic functions

Banach space valued analytic functions

The classical notions of

complex analytic function

,

holomorphic function

and

contour integral

of a

complex function

are easily generalized to

functions

f

:

ℂ

⟶

X

taking values on a

complex

Banach space

X

.

Moreover, the classical theory of complex analytic functions can still be applied, with suitable adjustments, to Banach space valued functions. In this way, important

theorems

such as Liouville’s theorem remain valid under this

generalization

.

In this entry we provide the definitions of analyticity and holomorphicity for Banach space valued functions, we give a definition of countour for this type of functions and discuss some useful results which enable the generalization of the classical theory.

0.1

Analiticity

Let

Ω

⊆

ℂ

be an open set and

X

a complex Banach space.

A function

f

:

Ω

⟶

X

is said to be

analytic

if each point

λ

0

∈

Ω

has a neighborhood in which

f

is the uniform limit of a

power series

with

coefficients

in

X

centered in

λ

0

f

⁢

(

λ

)

=

∑

k

=

0

∞

a

k

⁢

(

λ

-

λ

0

)

k

,

a

k

∈

X

Abel’s theorem on power series is still applicable changing

absolute values

|

.

|

by

vector norms

∥

.

∥

when appropriate.

0.2

Holomorphicity

A function

f

:

Ω

⟶

X

is said to be

differentiable

at a point

λ

0

∈

Ω

if the following limit exists (as a limit in

X

)

f

′

⁢

(

λ

0

)

:=

lim

λ

→

λ

0

⁡

f

⁢

(

λ

)

-

f

⁢

(

λ

0

)

λ

-

λ

0

f

is said to be in

S

⊂

Ω

if it is differentiable in a neighborhood of

S

.

The following Lemma is usefull in the generalization of the classical theory of holomorphic functions.

Lemma 1 -

Let

f

:

Ω

⟶

X

be a differentiable function at

λ

0

∈

Ω

. Let

ϕ

:

X

⟶

ℂ

be a

continuous

linear functional

in

X

. Then

ϕ

∘

f

:

Ω

⟶

ℂ

is differentiable at

λ

0

(in the classical sense) and

(

ϕ

∘

f

)

′

⁢

(

λ

0

)

=

ϕ

⁢

(

f

′

⁢

(

λ

0

)

)

Proof :

(

ϕ

∘

f

)

′

⁢

(

λ

0

)

=

lim

λ

→

λ

0

⁡

ϕ

⁢

(

f

⁢

(

λ

)

)

-

ϕ

⁢

(

f

⁢

(

λ

0

)

)

λ

-

λ

0

=

lim

λ

→

λ

0

⁡

ϕ

⁢

(

f

⁢

(

λ

)

-

f

⁢

(

λ

0

)

λ

-

λ

0

)

=

ϕ

⁢

(

lim

λ

→

λ

0

⁡

f

⁢

(

λ

)

-

f

⁢

(

λ

0

)

λ

-

λ

0

)

=

ϕ

⁢

(

f

′

⁢

(

λ

0

)

)

□

0.3

Contour Integrals

The usual way to relate the theory of complex analytic functions with the theory of holomorphic functions is by the use contour integrals. It is not different for Banach space valued functions.

We will define contour integrals for continuous Banach space valued functions but there’s no particular reason, besides the simplicity of , for restricting to this type of functions.

Let

γ

:

[

a

,

b

]

⟶

ℂ

be a piecewise smooth path in

Ω

⊆

ℂ

.
Let

f

:

Ω

⟶

X

be a

continuous function

. Let

𝒫

=

{

t

0

,

t

1

,

…

,

t

n

}

be a

partition

of

[

a

,

b

]

.

We define the

R

γ

⁢

(

f

,

𝒫

)

:=

∑

k

=

1

n

f

⁢

(

γ

⁢

(

t

k

)

)

⁢

(

γ

⁢

(

t

k

)

-

γ

⁢

(

t

k

-

1

)

)

and the of a partition

𝒫

as

∥

𝒫

∥

:=

max

k

⁡

|

t

k

-

t

k

-

1

|

The

contour integral

of

f

along

γ

is the element of

X

defined by

∫

γ

f

⁢

(

λ

)

⁢

𝑑

λ

:=

lim

∥

𝒫

∥

→

0

⁡

R

γ

⁢

(

f

,

𝒫

)

It can be shown that this limit always exists for continuous functions

f

.

The following Lemma is also usefull

Lemma 2 -

Let

γ

and

f

be as above. Let

ϕ

:

⟶

ℂ

be a continuous linear functional in

X

. Then

ϕ

⁢

(

∫

γ

f

⁢

(

λ

)

⁢

𝑑

λ

)

=

∫

γ

ϕ

∘

f

⁢

(

λ

)

⁢

𝑑

λ

Proof -

ϕ

⁢

(

∫

γ

f

⁢

(

λ

)

⁢

𝑑

λ

)

=

ϕ

⁢

(

lim

∥

𝒫

∥

→

0

⁡

R

γ

⁢

(

f

,

𝒫

)

)

=

ϕ

⁢

(

lim

∥

𝒫

∥

→

0

⁡

∑

k

=

1

n

f

⁢

(

γ

⁢

(

t

k

)

)

⁢

(

γ

⁢

(

t

k

)

-

γ

⁢

(

t

k

-

1

)

)

)

=

lim

∥

𝒫

∥

→

0

⁡

ϕ

⁢

(

∑

k

=

1

n

f

⁢

(

γ

⁢

(

t

k

)

)

⁢

(

γ

⁢

(

t

k

)

-

γ

⁢

(

t

k

-

1

)

)

)

=

lim

∥

𝒫

∥

→

0

⁡

∑

k

=

1

n

ϕ

⁢

(

f

⁢

(

γ

⁢

(

t

k

)

)

)

⁢

(

γ

⁢

(

t

k

)

-

γ

⁢

(

t

k

-

1

)

)

=

∫

γ

ϕ

∘

f

⁢

(

λ

)

⁢

𝑑

λ

⁢

□

0.4

Remarks

We have seen how the classical definitions generalize in straightforward way to Banach space valued functions. In fact, as we said before, the whole classical theory remains valid with proper adjustments.

As a example, we will prove a well-known theorem in

complex analysis

this time for Banach space valued functions.

Theorem -

Let

f

:

Ω

⟶

X

a continuous function with

antiderivative

F

. Let

γ

:

[

a

,

b

]

⟶

Ω

be a piecewise smooth path. Then

∫

γ

f

⁢

(

λ

)

⁢

𝑑

λ

=

F

⁢

(

γ

⁢

(

b

)

)

-

F

⁢

(

γ

⁢

(

a

)

)

Proof :

Let

ϕ

:

X

⟶

ℂ

be a continuous linear functional. Using Lemmas 1 and 2

ϕ

⁢

(

∫

γ

f

⁢

(

λ

)

⁢

𝑑

λ

)

=

∫

γ

ϕ

∘

f

⁢

(

λ

)

⁢

𝑑

λ

=

∫

γ

ϕ

∘

F

′

⁢

(

λ

)

⁢

𝑑

λ

=

∫

γ

(

ϕ

∘

F

)

′

⁢

(

λ

)

⁢

𝑑

λ

(

ϕ

∘

F

)

′

is a continuous function

Ω

⟶

ℂ

. As we know, this theorem is valued for complex valued functions. Then

∫

γ

(

ϕ

∘

F

)

′

⁢

(

λ

)

⁢

𝑑

λ

=

(

ϕ

∘

F

)

⁢

(

γ

⁢

(

b

)

)

-

(

ϕ

∘

F

)

⁢

(

γ

⁢

(

a

)

)

=

ϕ

⁢

[

F

⁢

(

γ

⁢

(

b

)

)

-

F

⁢

(

γ

⁢

(

a

)

)

]

Therefore

ϕ

⁢

(

∫

γ

f

⁢

(

λ

)

⁢

𝑑

λ

-

(

F

⁢

(

γ

⁢

(

b

)

)

-

F

⁢

(

γ

⁢

(

a

)

)

)

)

=

0

∀

ϕ

∈

X

′

As

X

is a Banach space, its

dual space

X

′

separates points (

http://planetmath.org/DualSpaceSeparatesPoints

), so we must have

∫

γ

f

⁢

(

λ

)

⁢

𝑑

λ

-

(

F

⁢

(

γ

⁢

(

b

)

)

-

F

⁢

(

γ

⁢

(

a

)

)

)

=

0

i.e.

∫

γ

f

⁢

(

λ

)

⁢

𝑑

λ

=

F

⁢

(

γ

⁢

(

b

)

)

-

F

⁢

(

γ

⁢

(

a

)

)

⁢

□

Title

Banach space valued analytic functions

Canonical name

BanachSpaceValuedAnalyticFunctions

Date of creation

2013-03-22 17:29:33

Last modified on

2013-03-22 17:29:33

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

10

Author

asteroid (17536)

Entry type

Feature

Classification

msc 46G20

Classification

msc 46G12

Classification

msc 46G10

Classification

msc 30G30

Classification

msc 47A56

Synonym

Banach space valued holomorphic function

Synonym

analytic Banach space valued function

Synonym

holomorphic Banach space valued function

Defines

contour integral of Banach space valued functions

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