aarjaneiro/planetmath huggingface dataset:
Schur’s Test

Schur’s Test

Theorem 1

.

(Schur’s Test)

Let

(

X

,

μ

)

be a

measure space

(

μ

a

positive measure

). Let

K

be a positive,

measurable function

on

X

×

X

. Define the

operator

T

⁢

f

⁢

(

x

)

:=

∫

X

K

⁢

(

x

,

y

)

⁢

f

⁢

(

y

)

⁢

𝑑

μ

⁢

(

y

)

,

x

∈

X

If for some

1

<

p

<

∞

there exists a

measurable

, strictly positive

function

h

and a

constant

M

>

0

such that

∫

X

K

⁢

(

x

,

y

)

⁢

h

⁢

(

y

)

q

⁢

𝑑

μ

⁢

(

y

)

≤

M

⁢

h

⁢

(

x

)

q

∫

X

K

⁢

(

x

,

y

)

⁢

h

⁢

(

x

)

p

⁢

𝑑

μ

⁢

(

x

)

≤

M

⁢

h

⁢

(

y

)

p

with

p

-

1

+

q

-

1

=

1

, then

||

T

||

≤

M

in

L

p

⁢

(

X

,

d

⁢

μ

)

.

Proof.

Let

f

∈

L

p

⁢

(

X

,

d

⁢

μ

)

. We have

|

T

⁢

f

⁢

(

x

)

|

≤

∫

X

h

⁢

(

y

)

⁢

h

⁢

(

y

)

-

1

⁢

|

f

⁢

(

y

)

|

⁢

K

⁢

(

x

,

y

)

⁢

𝑑

μ

⁢

(

y

)

hence by Hoelder’s

inequality

|

T

⁢

f

⁢

(

x

)

|

≤

[

∫

X

K

⁢

(

x

,

y

)

⁢

h

⁢

(

y

)

q

⁢

𝑑

μ

⁢

(

y

)

]

1

q

⁢

[

∫

X

K

⁢

(

x

,

y

)

⁢

h

⁢

(

y

)

-

p

⁢

|

f

⁢

(

y

)

|

p

⁢

𝑑

μ

⁢

(

y

)

]

1

p

By the first inequality in the

assumption

we have

|

T

⁢

f

⁢

(

x

)

|

≤

M

1

q

⁢

h

⁢

(

x

)

⁢

[

∫

X

K

⁢

(

x

,

y

)

⁢

h

⁢

(

y

)

-

p

⁢

|

f

⁢

(

y

)

|

p

⁢

𝑑

μ

⁢

(

y

)

]

1

p

Evaluating

||

T

⁢

f

||

p

p

by Fubini and the second inequality in the assumption we obtain

∫

X

|

T

⁢

f

⁢

(

x

)

|

p

⁢

𝑑

μ

⁢

(

x

)

≤

M

p

⁢

∫

X

|

f

⁢

(

y

)

|

p

⁢

𝑑

μ

⁢

(

y

)

This

completes

the proof.
∎

A noted special case is Young’s Inequality

Corollary 1

.

(Young)

Let

K

:

R

n

×

R

n

→

C

be Borel-measurable such that there is a constant

C

>

0

with

sup

x

∈

ℝ

n

⁡

∫

ℝ

n

|

K

⁢

(

x

,

y

)

|

⁢

𝑑

λ

n

⁢

(

y

)

≤

C

sup

y

∈

ℝ

n

⁡

∫

ℝ

n

|

K

⁢

(

x

,

y

)

|

⁢

𝑑

λ

n

⁢

(

x

)

≤

C

For

f

∈

L

p

⁢

(

R

n

)

(

1

≤

p

≤

+

∞

) define

T

⁢

(

f

)

⁢

(

x

)

:=

∫

ℝ

n

K

⁢

(

x

,

y

)

⁢

f

⁢

(

y

)

⁢

𝑑

λ

n

⁢

(

y

)

Then

||

T

⁢

f

||

p

≤

C

⁢

||

f

||

p

.

References

(Hedenmalm 2000)

H. Hedenmalm, Boris Korenblum, Kehe Zhu

Theory of Bergman spaces

, Springer Verlag, New York, 2000

Title

Schur’s Test

Canonical name

SchursTest

Date of creation

2013-03-22 19:01:19

Last modified on

2013-03-22 19:01:19

Owner

karstenb (16623)

Last modified by

karstenb (16623)

Numerical id

6

Author

karstenb (16623)

Entry type

Theorem

Classification

msc 46G99

Generated on Fri Feb 9 21:41:00 2018 by

LaTeXML
approximate identity

approximate identity

Let

𝒜

be a

Banach algebra

.

A

left approximate identity

for

𝒜

is a net

(

e

λ

)

λ

∈

Λ

in

𝒜

which :

1.

∥

e

λ

∥

<

C

∀

λ

∈

Λ

, for some constant

C

.

2.

e

λ

⁢

a

⟶

a

, for every

a

∈

𝒜

.

Similarly, a

right approximate identity

for

𝒜

is a net

(

e

λ

)

λ

∈

Λ

in

𝒜

which :

1.

∥

e

λ

∥

<

C

∀

λ

∈

Λ

, for some constant

C

.

2.

a

⁢

e

λ

⟶

a

, for every

a

∈

𝒜

.

An

approximate identity

for a

𝒜

is a net

(

e

λ

)

λ

∈

Λ

in

𝒜

which is both a left and right approximate identity.

0.0.1

Remarks:

•

There are examples of Banach algebras that do not have approximate .

•

If

𝒜

has an

identity element

e

, then clearly

e

itself is an approximate identity for

𝒜

.

Title

approximate identity

Canonical name

ApproximateIdentity

Date of creation

2013-03-22 17:30:22

Last modified on

2013-03-22 17:30:22

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46H05

Synonym

approximate unit

Defines

left approximate identity

Defines

right approximate identity

Generated on Fri Feb 9 21:41:02 2018 by

LaTeXML
Banach algebra

Banach algebra

Definition 1

.

A

Banach algebra

𝒜

is a

Banach space

(over

ℂ

) with an multiplication law compatible with the norm which turns

𝒜

into an algebra. Compatibility with the norm means that, for all

a

,

b

∈

𝒜

, it is the case that the following

product

inequality

holds:

∥

a

⁢

b

∥

≤

∥

a

∥

⁢

∥

b

∥

Definition 2

.

A

Banach *-algebra

is a Banach algebra

𝒜

with a map

:

*

𝒜

→

𝒜

which satisfies the following properties:

a

*

*

=

a

,

(1)

(

a

⁢

b

)

*

=

b

*

⁢

a

*

,

(2)

(

a

+

b

)

*

=

a

*

+

b

*

,

(3)

(

λ

⁢

a

)

*

=

λ

¯

⁢

a

*

∀

λ

∈

ℂ

,

(4)

∥

a

*

∥

=

∥

a

∥

,

(5)

where

λ

¯

is the

complex conjugation

of

λ

. In other words, the

operator

*

is an

involution

.

Example 1

The algebra of

bounded operators

on a Banach space is a Banach algebra
for the

operator norm

.

Title

Banach algebra

Canonical name

BanachAlgebra

Date of creation

2013-03-22 12:57:52

Last modified on

2013-03-22 12:57:52

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 46H05

Synonym

B-algebra

Synonym

Banach *-algebra

Synonym

B*-algebra

Synonym

B

*

-algebra

Related topic

ExampleOfLinearInvolution

Related topic

GelfandTornheimTheorem

Related topic

MultiplicativeLinearFunctional

Related topic

TopologicalAlgebra

Generated on Fri Feb 9 21:41:04 2018 by

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Banach-Krein-Šmulian theorem

Banach-Krein-Šmulian theorem

Let

E

be a

Banach space

. A

convex subset

C

of the

dual
space

E

*

is closed in the weak-

*

topology

if and only
if the

intersection

of

C

with the
ball

B

r

⁢

(

0

)

is weak-

*

closed for every

r

>

0

.

References

1

Dunford, N., and J. T. Schwartz,

Linear Operators

,
Part I, Interscience Publishers, 1967.

Title

Banach-Krein-Šmulian

theorem

Canonical name

BanachKreinvSmulianTheorem

Date of creation

2013-03-22 15:14:52

Last modified on

2013-03-22 15:14:52

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

21

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 46H05

Generated on Fri Feb 9 21:41:06 2018 by

LaTeXML
Gelfand-Mazur theorem

Gelfand-Mazur theorem

Theorem -

Let

𝒜

be a unital

Banach algebra

over

ℂ

that is also a

division algebra

(i.e. every non-zero element is

invertible

). Then

𝒜

is

isometrically isomorphic

to

ℂ

.

Proof :

Let

e

denote the unit of

𝒜

.

Let

x

∈

𝒜

and

σ

⁢

(

x

)

be its spectrum. It is known that the spectrum is a non-empty set (

http://planetmath.org/SpectrumIsANonEmptyCompactSet

) in

ℂ

.

Let

λ

∈

σ

⁢

(

x

)

. Since

x

-

λ

⁢

e

is not invertible and

𝒜

is a division algebra, we must have

x

-

λ

⁢

e

=

0

and so

x

=

λ

⁢

e

Let

ϕ

:

ℂ

⟶

𝒜

be defined by

ϕ

⁢

(

λ

)

=

λ

⁢

e

.

It is clear that

ϕ

is an injective algebra homomorphism.

By the above discussion,

ϕ

is also surjective.

It is isometric because

∥

λ

⁢

e

∥

=

|

λ

|

⁢

∥

e

∥

=

|

λ

|

Therefore,

𝒜

is isometrically isomorphic to

ℂ

.

□

Title

Gelfand-Mazur theorem

Canonical name

GelfandMazurTheorem

Date of creation

2013-03-22 17:29:03

Last modified on

2013-03-22 17:29:03

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46H05

Generated on Fri Feb 9 21:41:08 2018 by

LaTeXML
Hahn-Banach theorem (geometric form)

Hahn-Banach theorem (geometric form)

Let

E

be a locally convex space

A

,

B

⊂

E

, two convex, nonempty and disjoint sets. Suppose that

A

is closed and

B

is

compact

. Then there is a

hyperplane

which

strictly

separates

A

and

B

.

Title

Hahn-Banach theorem

(

geometric form

)

Canonical

name

HahnBanachTheoremgeometricForm

Date of creation

2013-03-22 15:15:09

Last modified on

2013-03-22 15:15:09

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

8

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 46H05

Related topic

HyperplaneSeparation

Generated on Fri Feb 9 21:41:09 2018 by

LaTeXML
Hilbert spaces are uniformly convex spaces

Hilbert spaces are uniformly convex spaces

Theorem 1

.

A

Hilbert space

is a

uniformly convex space

.

Title

Hilbert spaces are uniformly convex spaces

Canonical name

HilbertSpacesAreUniformlyConvexSpaces

Date of creation

2013-03-22 15:14:37

Last modified on

2013-03-22 15:14:37

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

14

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 46H05

Classification

msc 46C15

Generated on Fri Feb 9 21:41:11 2018 by

LaTeXML
hyperplane

hyperplane

Let

E

be a

linear space

over a field

k

. A

hyperplane

H

in

E

is defined as the set of the form

H

=

{

x

∈

E

:

f

⁢

(

x

)

=

a

}

where

a

∈

k

and

f

is a nonzero

linear functional

,

f

:

E

→

k

. If

k

=

ℝ

or

ℂ

, then

H

is called a

real hyperplane

or

complex hyperplane

respectively.

Remark

. When

k

=

ℂ

, the word “hyperplane” also has a more restrictive meaning: it is the zero set of a complex linear functional (by setting

a

=

0

above).

Title

hyperplane

Canonical

name

Hyperplane

Date of creation

2013-03-22 15:15:12

Last modified on

2013-03-22 15:15:12

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

9

Author

georgiosl (7242)

Entry type

Definition

Classification

msc 46H05

Defines

real hyperplane

Defines

complex hyperplane

Generated on Fri Feb 9 21:41:13 2018 by

LaTeXML
invertible elements in a Banach algebra form an open set

invertible elements in a Banach algebra form an open set

Theorem -

Let

𝒜

be a Banach algebra with

identity element

e

and

G

⁢

(

𝒜

)

be the set of invertible elements in

𝒜

. Let

B

r

⁢

(

x

)

denote the open ball of radius

r

centered in

x

.

Then, for all

x

∈

G

⁢

(

𝒜

)

we have that

B

∥

x

-

1

∥

-

1

⁢

(

x

)

⊆

G

⁢

(

𝒜

)

and therefore

G

⁢

(

𝒜

)

is open in

𝒜

.

Proof :

Let

x

∈

G

⁢

(

𝒜

)

and

y

∈

B

∥

x

-

1

∥

-

1

⁢

(

x

)

. We have that

∥

e

-

x

-

1

⁢

y

∥

=

∥

x

-

1

⁢

x

-

x

-

1

⁢

y

∥

=

∥

x

-

1

⁢

(

x

-

y

)

∥

≤

∥

x

-

1

∥

⁢

∥

x

-

y

∥

<

∥

x

-

1

∥

⁢

∥

x

-

1

∥

-

1

=

1

So, by the

Neumann series

(

http://planetmath.org/NeumannSeriesInBanachAlgebras

) we conclude that

e

-

(

e

-

x

-

1

⁢

y

)

is

invertible

,
i.e.

x

-

1

⁢

y

∈

G

⁢

(

𝒜

)

.

As

G

⁢

(

𝒜

)

is a group we must have

y

∈

G

⁢

(

𝒜

)

.

So

B

∥

x

-

1

∥

-

1

⁢

(

x

)

⊆

G

⁢

(

𝒜

)

and the theorem follows.

□

Title

invertible elements in a Banach algebra form an open set

Canonical name

InvertibleElementsInABanachAlgebraFormAnOpenSet

Date of creation

2013-03-22 17:23:22

Last modified on

2013-03-22 17:23:22

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46H05

Generated on Fri Feb 9 21:41:15 2018 by

LaTeXML
multiplicative linear functional

multiplicative linear functional

1

Definition

Let

𝒜

be an algebra over

ℂ

.

A

multiplicative linear functional

is an nontrivial algebra homomorphism

ϕ

:

𝒜

⟶

ℂ

, i.e.

ϕ

is a non-zero

linear functional

such that

ϕ

⁢

(

x

⋅

y

)

=

ϕ

⁢

(

x

)

⋅

ϕ

⁢

(

y

)

,

∀

x

,

y

∈

𝒜

.

Multiplicative linear functionals are also called

characters

of

𝒜

.

2

Properties

•

If

ϕ

is a multiplicative linear functional in a

Banach algebra

𝒜

over

ℂ

then

ϕ

is

continuous

. Moreover, if

𝒜

has an

identity element

then

∥

ϕ

∥

=

1

.

•

Suppose

𝒜

is a Banach algebra over

ℂ

. The set of multiplicative linear functionals in

𝒜

is a

locally compact Hausdorff space

in the weak-*

topology

. Moreover, this set is

compact

if

𝒜

has an identity element.

•

Suppose

𝒜

is a commutative Banach algebra over

ℂ

with an identity element. There is a bijective correspondence
between the set of

maximal ideals

in

𝒜

and the set of multiplicative linear functionals
in

𝒜

. This correspondence is given by

ϕ

⟼

K

⁢

e

⁢

r

⁢

ϕ

•

Suppose

𝒜

is a

commutative

C

*

-

algebra

(

http://planetmath.org/CAlgebra

). Multiplicative linear functionals in

𝒜

are exactly the

irreducible representations

(

http://planetmath.org/BanachAlgebraRepresentation

) of

𝒜

.

3

Character space of a Banach algebra

As stated above, the set of all multiplicative linear functionals in a Banach algebra

𝒜

is a locally compact Hausdorff space with the weak-* topology. It becomes a compact set if

𝒜

has an identity element.

There are several designations for this space, such as:
the of

𝒜

, the

maximal ideal space

, the

character space

.

4

Examples

•

Let

X

be a

topological space

and

C

⁢

(

X

)

the algebra of continuous functions

X

⟶

ℂ

. Every point evaluation is a multiplicative linear functional of

C

⁢

(

X

)

. In other words, for every point

x

∈

X

, the function

e

⁢

v

x

:

C

⁢

(

X

)

⟶

ℂ

e

⁢

v

x

⁢

(

f

)

=

f

⁢

(

x

)

that gives the evaluation in

x

, is a multiplicative linear functional of

C

⁢

(

X

)

.

Title

multiplicative linear functional

Canonical name

MultiplicativeLinearFunctional

Date of creation

2013-03-22 17:22:25

Last modified on

2013-03-22 17:22:25

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

29

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46H05

Synonym

character (of an algebra)

Related topic

LinearFunctional

Related topic

GelfandTransform

Related topic

BanachAlgebra

Defines

character

Defines

maximal ideal space

Defines

character space

Generated on Fri Feb 9 21:41:17 2018 by

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Neumann series in Banach algebras

Neumann series in Banach algebras

The

Neumann series

can be generalized to a

Banach algebra

𝒜

with

identity element

e

.
Thus we have:

If

x

∈

𝒜

is such that

∥

x

∥

<

1

then

e

-

x

is

invertible

with

inverse

given by

(

e

-

x

)

-

1

=

∑

n

=

0

∞

x

n

and

∥

(

e

-

x

)

-

1

∥

≤

1

1

-

∥

x

∥

Title

Neumann series in Banach algebras

Canonical name

NeumannSeriesInBanachAlgebras

Date of creation

2013-03-22 17:23:19

Last modified on

2013-03-22 17:23:19

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46H05

Generated on Fri Feb 9 21:41:19 2018 by

LaTeXML
normed algebra

normed algebra

A ring

A

is said to be a

normed ring

if

A

possesses a norm

∥

⋅

∥

, that is, a non-negative real-valued function

∥

⋅

∥

:

A

→

ℝ

such that for any

a

,

b

∈

A

,

1.

∥

a

∥

=

0

iff

a

=

0

,

2.

∥

a

+

b

∥

≤

∥

a

∥

+

∥

b

∥

,

3.

∥

-

a

∥

=

∥

a

∥

, and

4.

∥

a

⁢

b

∥

≤

∥

a

∥

⁢

∥

b

∥

.

Remarks

.

•

If

A

contains the

multiplicative identity

1

, then

0

<

∥

1

∥

≤

∥

1

∥

⁢

∥

1

∥

and so

1

≤

∥

1

∥

.

•

However, it is usually required that in a normed ring,

∥

1

∥

=

1

.

•

∥

⋅

∥

defines a metric

d

on

A

given by

d

⁢

(

a

,

b

)

=

∥

a

-

b

∥

, so that

A

with

d

is a

metric space

and one can set up a

topology

on

A

by defining its

subbasis

a collection of

B

⁢

(

a

,

r

)

:=

{

x

∈

A

∣

d

⁢

(

a

,

x

)

<

r

}

called

open balls

for any

a

∈

A

and

r

>

0

. With this definition, it is easy to see that

∥

⋅

∥

is

continuous

.

•

Given a sequence

{

a

n

}

of elements in

A

, we say that

a

is a

limit point

of

{

a

n

}

, if

lim

n

→

∞

⁡

∥

a

n

-

a

∥

=

0

.

By the

triangle inequality

,

a

, if it exists, is unique, and so we also write

a

=

lim

n

→

∞

⁡

a

n

.

•

In addition, the last condition ensures that the ring multiplication is continuous.

An

algebra

A

over a field

k

is said to be a

normed algebra

if

1.

A

is a normed ring with norm

∥

⋅

∥

,

2.

k

is equipped with a

valuation

|

⋅

|

, and

3.

∥

α

⁢

a

∥

=

|

α

|

⁢

∥

a

∥

for any

α

∈

k

and

a

∈

A

.

Remarks

.

•

Alternatively, a normed algebra

A

can be defined as a

normed vector space

with a multiplication defined on

A

such that multiplication is continuous with respect to the norm

∥

⋅

∥

.

•

Typically,

k

is either the reals

ℝ

or the

complex numbers

ℂ

, and

A

is called a

real normed algebra

or a

complex normed algebra

correspondingly.

•

A normed algebra that is

complete

with respect to the norm is called

Banach algebra

(the underlying field must be complete and

algebraically closed

), paralleling with the analogy with a

Banach space

versus a normed vector space.

•

Normed rings and normed algebras are special cases of the more general notions of a

topological ring

and a

topological algebra

, the latter of which is defined as a topological ring over a field such that the scalar multiplication is continuous.

References

1

M. A. Naimark:

Normed Rings

, Noordhoff, (1959).

2

C. E. Rickart:

General Theory of Banach Algebras

, Van Nostrand, 1960.

Title

normed algebra

Canonical name

NormedAlgebra

Date of creation

2013-03-22 16:11:38

Last modified on

2013-03-22 16:11:38

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 46H05

Related topic

GelfandTornheimTheorem

Related topic

SuperfieldsSuperspace

Defines

normed ring

Defines

topological algebra

Defines

real normed algebra

Defines

complex normed algebra

Generated on Fri Feb 9 21:41:29 2018 by

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proof of Neumann series in Banach algebras

proof of Neumann series in Banach algebras

Let x be an element of a

Banach algebra

with identity,

∥

x

∥

<

1

. By applying the properties of the Norm in a Banach algebra, we see that the partial sums form a

Cauchy sequence

:

∥

∑

n

=

l

m

x

n

∥

≤

∑

n

=

l

m

∥

x

∥

n

→

0

for

l

,

m

→

∞

(as is well known from

real analysis

), so by completeness of the Banach Algebra, the series

converges

to some element

y

=

∑

n

=

0

∞

x

n

.

We observe that for any

m

∈

ℕ

,

(

1

-

x

)

⁢

∑

n

=

0

m

x

n

=

∑

n

=

0

m

x

n

-

∑

n

=

1

m

+

1

x

n

=

1

-

x

m

+

1

(1)

Furthermore,

∥

x

m

+

1

∥

≤

∥

x

∥

m

+

1

, so

lim

m

⁡

x

m

+

1

=

0

.

Thus, by taking the limit

m

→

∞

on both sides of (1), we get

(

1

-

x

)

⁢

y

=

1

(We can exchange the limit with the multiplication by

(

1

-

x

)

, since the multiplication in Banach algebras is

continuous

)

Since the Banach algebra generated by a single element is

commutative

and

(

1

-

x

)

and

y

are both in the Banach algebra generated by

x

, we also get

y

⁢

(

1

-

x

)

=

1

. Hence,

y

=

(

1

-

x

)

-

1

.

As in the first paragraph, the last claim

y

≤

1

1

-

∥

y

∥

again follows by applying the geometric series for

real numbers

.

Title

proof of Neumann series in Banach algebras

Canonical name

ProofOfNeumannSeriesInBanachAlgebras

Date of creation

2013-03-22 17:32:40

Last modified on

2013-03-22 17:32:40

Owner

FunctorSalad (18100)

Last modified by

FunctorSalad (18100)

Numerical id

5

Author

FunctorSalad (18100)

Entry type

Proof

Classification

msc 46H05

Generated on Fri Feb 9 21:41:31 2018 by

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property of uniformly convex Banach Space

property of uniformly convex Banach Space

Theorem 1

.

Let

X

be a

uniformly convex

Banach space

. Let

(

x

n

)

be a sequence in

X

such that

lim

⁡

x

n

=

x

in the weak-topology

(

w

⁢

(

X

,

X

*

)

)

and

lim sup

⁡

∥

x

n

∥

≤

∥

x

∥

.Then

x

n

converges

to

x

.

Proof.

For

x

=

0

the claim is obvious, so suppose that

x

≠

0

.The sequence

(

x

n

)

n

≥

1

converges to

x

for

w

-

t

⁢

o

⁢

p

⁢

o

⁢

l

⁢

o

⁢

g

⁢

y

⇒

∥

x

∥

≤

lim inf

⁡

∥

x

n

∥

.So let

λ

n

=

max

⁡

{

∥

x

∥

,

∥

x

n

∥

}

and we have that

lim

⁡

λ

n

=

∥

x

∥

.
Define

y

n

=

x

n

λ

n

and

y

=

x

∥

x

∥

.Then

y

n

converges to

y

in

w

-

t

⁢

o

⁢

p

⁢

o

⁢

l

⁢

o

⁢

g

⁢

y

.
We conclude that

∥

y

∥

≤

lim inf

⁡

∥

y

n

+

y

2

∥

. Also,

∥

y

∥

=

1

,

∥

y

n

∥

≤

1

so we have that

lim

⁡

∥

y

n

+

y

2

∥

=

1

. As the Banach space is uniformly convex one can easily see that

lim

⁡

∥

y

n

-

y

∥

=

0

. Therefore

x

n

converges to

x

. The proof is

complete

.
∎

Title

property of uniformly convex Banach Space

Canonical name

PropertyOfUniformlyConvexBanachSpace

Date of creation

2013-03-22 15:14:18

Last modified on

2013-03-22 15:14:18

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

20

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 46H05

Generated on Fri Feb 9 21:41:33 2018 by

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resolvent function is analytic

resolvent function is analytic

Theorem

-

Let

𝒜

be a complex

Banach algebra

with

identity element

e

. Let

x

∈

𝒜

and

σ

⁢

(

x

)

denote its spectrum.

Then, the

resolvent

function

(

http://planetmath.org/ResolventMatrix

)

R

x

:

ℂ

-

σ

⁢

(

x

)

⟶

𝒜

defined by

R

x

⁢

(

λ

)

=

(

x

-

λ

⁢

e

)

-

1

is

analytic

(

http://planetmath.org/BanachSpaceValuedAnalyticFunctions

).

Moreover, for each

λ

0

∈

ℂ

-

σ

⁢

(

x

)

it has the power series

R

x

⁢

(

λ

)

=

∑

n

=

0

∞

R

x

⁢

(

λ

0

)

n

+

1

⁢

(

λ

-

λ

0

)

n

(1)

where the series converges absolutely for each

λ

in the open disk centered in

λ

0

given by

|

λ

-

λ

0

|

<

1

∥

R

x

⁢

(

λ

0

)

∥

(2)

Proof :

Analyticity is defined for functions whose domain is open.

Thus, we start by proving that

ℂ

-

σ

⁢

(

x

)

is an open set in

ℂ

. To do so it is enough to prove that for every

λ

0

∈

ℂ

-

σ

⁢

(

x

)

the open disk defined by (2) above is

contained

in

ℂ

-

σ

⁢

(

x

)

.

Let

λ

0

∈

ℂ

-

σ

⁢

(

x

)

and

λ

be such that

|

λ

-

λ

0

|

<

1

∥

R

x

⁢

(

λ

0

)

∥

Then

∥

(

λ

-

λ

0

)

⁢

R

x

⁢

(

λ

0

)

∥

<

1

and by the Neumann series (

http://planetmath.org/NeumannSeriesInBanachAlgebras

)

e

-

(

λ

-

λ

0

)

⁢

R

x

⁢

(

λ

0

)

is

invertible

.

Since

λ

0

∉

σ

⁢

(

x

)

it follows that

(

x

-

λ

0

⁢

e

)

is invertible.

Hence, from the equality

x

-

λ

⁢

e

=

x

-

λ

0

⁢

e

-

(

λ

-

λ

0

)

⁢

e

=

(

x

-

λ

0

⁢

e

)

⋅

[

e

-

(

λ

-

λ

0

)

⁢

R

x

⁢

(

λ

0

)

]

(3)

we conclude that

x

-

λ

⁢

e

is also invertible, i.e.

λ

∈

ℂ

-

σ

⁢

(

x

)

. Thus

ℂ

-

σ

⁢

(

x

)

is open.

The above proof also pointed out that for every

λ

0

∈

ℂ

,

R

x

is defined in the open disk of radius

1

∥

R

x

⁢

(

λ

0

)

∥

centered in

λ

0

.

We now prove the analyticity of the .

Taking

inverses

on the equality (3) above one obtains

R

x

⁢

(

λ

)

=

(

e

-

(

λ

-

λ

0

)

⁢

R

x

⁢

(

λ

0

)

)

-

1

⋅

R

x

⁢

(

λ

0

)

Again, by the Neumann series (

http://planetmath.org/NeumannSeriesInBanachAlgebras

), one obtains

R

x

⁢

(

λ

)

=

[

∑

n

=

0

∞

R

x

⁢

(

λ

0

)

n

⁢

(

λ

-

λ

0

)

n

]

⋅

R

x

⁢

(

λ

0

)

=

∑

n

=

0

∞

R

x

⁢

(

λ

0

)

n

+

1

⁢

(

λ

-

λ

0

)

n

□

Title

resolvent function is analytic

Canonical name

ResolventFunctionIsAnalytic

Date of creation

2013-03-22 17:29:36

Last modified on

2013-03-22 17:29:36

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

8

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46H05

Classification

msc 47A10

Generated on Fri Feb 9 21:41:36 2018 by

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