aarjaneiro/planetmath huggingface dataset:
rotund space

rotund space

A

normed space

is said to be

rotund

if every point of

C

⁢

(

0

,

1

)

is an extreme point. Here

C

⁢

(

0

,

1

)

is the set

{

b

:

∥

b

∥

=

1

}

.
Equivalently, a space is rotund if and only if

a

≠

b

and

∥

a

∥

=

∥

b

∥

≤

1

implies

∥

a

+

b

∥

<

2

.

A

uniformly convex space

is rotund.

Title

rotund space

Canonical

name

RotundSpace

Date of creation

2013-03-22 16:04:56

Last modified on

2013-03-22 16:04:56

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 46H05

Generated on Fri Feb 9 21:41:38 2018 by

LaTeXML
topological divisor of zero

topological divisor of zero

Let

A

be a

normed ring

.  An element

a

∈

A

is said to be a

left

topological divisor of zero

if there is a sequence

a

n

with

∥

a

n

∥

=

1

for all

n

such that

lim

n

→

∞

⁡

∥

a

⁢

a

n

∥

=

0

.

Analogously,

a

is a if

lim

n

→

∞

⁡

∥

b

n

⁢

a

∥

=

0

,

for some sequence

b

n

with

∥

b

n

∥

=

1

.  The element

a

is a

topological divisor of zero

if it is both a left and a topological divisor of zero.

Remarks

.

•

Any

zero divisor

is a topological divisor of zero.

•

If

a

is a (left) topological divisor of zero, then

b

⁢

a

is a (left) topological divisor of zero. As a result,

a

is never a unit, for if

b

is its

inverse

, then

1

=

b

⁢

a

would be a topological divisor of zero, which is impossible.

•

In a commutative Banach algebra

A

, an element is a topological divisor of zero if it lies on the boundary of

U

⁢

(

A

)

, the

group of units

of

A

.

Title

topological divisor of zero

Canonical

name

TopologicalDivisorOfZero

Date of creation

2013-03-22 16:12:15

Last modified on

2013-03-22 16:12:15

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 46H05

Synonym

generalized divisor of zero

Generated on Fri Feb 9 21:41:50 2018 by

LaTeXML
topologically nilpotent

topologically nilpotent

An element

a

in a

normed ring

A

is said to be

topologically nilpotent

if

lim

n

→

∞

⁡

∥

a

n

∥

1

n

=

0

.

Topologically nilpotent elements are also called

quasinilpotent

.

Remarks

.

•

Any

nilpotent element

is topologically nilpotent.

•

If

a

and

b

are topologically nilpotent and

a

⁢

b

=

b

⁢

a

, then

a

⁢

b

is topologically nilpotent.

•

When

A

is a unital

Banach algebra

, an element

a

∈

A

is topologically nilpotent iff its spectrum

σ

⁢

(

a

)

equals

{

0

}

.

Title

topologically nilpotent

Canonical name

TopologicallyNilpotent

Date of creation

2013-03-22 16:12:04

Last modified on

2013-03-22 16:12:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 46H05

Synonym

quasinilpotent

Generated on Fri Feb 9 21:42:00 2018 by

LaTeXML
uniformly convex Banach space is reflexive

uniformly convex Banach space is reflexive

Every

uniformly convex Banach space
is reflexive

(

http://planetmath.org/DualSpace

).

Title

uniformly convex Banach space is reflexive

Canonical name

UniformlyConvexBanachSpaceIsReflexive

Date of creation

2013-03-22 15:14:58

Last modified on

2013-03-22 15:14:58

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

18

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 46H05

Generated on Fri Feb 9 21:42:01 2018 by

LaTeXML
uniformly convex space

uniformly convex space

A

normed space

is

uniformly convex

iff

∀

ϵ

>

0

there exists

δ

>

0

that satisfies
for

∥

x

∥

≤

1

∥

y

∥

≤

1

and

∥

x

-

y

∥

>

ϵ

⇒

∥

x

+

y

2

∥

≤

1

-

δ

.

For example it is easily seen that the normed space

(

ℝ

2

,

∥

.

∥

2

)

is uniformly convex space.
Also

L

p

and

l

p

spaces for

1

<

p

<

∞

are uniformly convex, see

J.A. Clarkson, ”Uniformly convex spaces”, Trans. Amer. Math. Society, 40 (1936), 396-414.

Title

uniformly convex space

Canonical

name

UniformlyConvexSpace

Date of creation

2013-03-22 15:13:11

Last modified on

2013-03-22 15:13:11

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

32

Author

georgiosl (7242)

Entry type

Definition

Classification

msc 46H05

Synonym

uniformly convex

Generated on Fri Feb 9 21:42:03 2018 by

LaTeXML
quotients of Banach algebras

quotients of Banach algebras

Theorem -

Let

𝒜

be a

Banach algebra

and

ℐ

⊆

𝒜

a closed (

http://planetmath.org/ClosedSet

) ideal. Then

𝒜

/

ℐ

is Banach algebra under the

quotient norm

.

Proof:

Denote the quotient norm by

∥

⋅

∥

q

.

By the parent entry (

http://planetmath.org/QuotientsOfBanachSpacesByClosedSubspacesAreBanachSpacesUnderTheQuotientNorm

) we know that

𝒜

/

ℐ

is a

Banach space

under the quotient norm. Thus, we only need to show the

normed algebra

inequality

:

∥

a

⁢

b

+

ℐ

∥

q

≤

∥

a

+

ℐ

∥

q

⁢

∥

b

+

ℐ

∥

q

for every

a

,

b

∈

𝒜

.

Using the fact that

𝒜

is a Banach algebra and the definition of quotient norm we have that:

∥

a

⁢

b

+

ℐ

∥

q

=

inf

z

∈

a

⁢

b

+

ℐ

⁡

∥

z

∥

=

inf

u

∈

a

+

ℐ

v

∈

b

+

ℐ

⁡

∥

u

⁢

v

∥

≤

inf

u

∈

a

+

ℐ

v

∈

b

+

ℐ

⁡

∥

u

∥

⁢

∥

v

∥

≤

inf

u

∈

a

+

ℐ

⁡

∥

u

∥

⁢

inf

v

∈

b

+

ℐ

⁡

∥

v

∥

=

∥

a

+

ℐ

∥

q

⁢

∥

b

+

ℐ

∥

q

□

Title

quotients of Banach algebras

Canonical name

QuotientsOfBanachAlgebras

Date of creation

2013-03-22 17:41:54

Last modified on

2013-03-22 17:41:54

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46H10

Classification

msc 46H05

Classification

msc 46B99

Generated on Fri Feb 9 21:42:05 2018 by

LaTeXML
spectral invariance theorem (for C^*-algebras)

spectral invariance theorem (for

C

*

-algebras)

The

spectral permanence theorem

( entry) relates the spectrums

σ

ℬ

⁢

(

x

)

and

σ

𝒜

⁢

(

x

)

of an

element

x

∈

ℬ

⊆

𝒜

relatively to the

Banach algebras

ℬ

and

𝒜

.

For

C

*

-

algebras

(

http://planetmath.org/CAlgebra

) the situation is quite .

Spectral invariance theorem

-

Suppose

𝒜

is a unital

C

*

-algebra and

ℬ

⊆

𝒜

a

C

*

-

subalgebra

that contains the

identity

of

𝒜

. Then for every

x

∈

ℬ

one has

σ

ℬ

⁢

(

x

)

=

σ

𝒜

⁢

(

x

)

.

The spectral invariance theorem is a straightforward corollary of the next more general

theorem

about invertible elements in

C

*

-subalgebras.

Theorem -

Let

x

∈

ℬ

⊂

𝒜

be as above. Then

x

is

invertible

in

ℬ

if and only if

x

invertible in

𝒜

.

Proof :

•

(

⟹

)

If

x

is invertible in

ℬ

then it is clearly invertible in

𝒜

.

•

(

⟸

)

If

x

is invertible in

𝒜

, then so is

y

=

x

*

⁢

x

. Thus,

0

∉

σ

𝒜

⁢

(

y

)

.

Since

y

is self-adjoint (

http://planetmath.org/InvolutaryRing

),

σ

𝒜

⁢

(

y

)

⊆

ℝ

(see this entry (

http://planetmath.org/SpecialElementsInACAlgebraAndTheirSpectralProperties

)), and so

ℂ

-

σ

𝒜

⁢

(

y

)

has no

bounded

(

http://planetmath.org/Bounded

)

connected components

.

By the spectral permanence theorem (

http://planetmath.org/SpectralPermanenceTheorem

) we must have

σ

ℬ

⁢

(

y

)

=

σ

𝒜

⁢

(

y

)

. Hence,

0

∉

σ

ℬ

⁢

(

y

)

, i.e.

y

is invertible in

ℬ

.

It follows that

x

-

1

=

(

x

*

⁢

x

)

-

1

⁢

x

*

=

y

-

1

⁢

x

*

∈

ℬ

, i.e.

x

is invertible in

ℬ

.

□

Title

spectral invariance theorem (for

C

*

-algebras)

Canonical name

SpectralInvarianceTheoremforCalgebras

Date of creation

2013-03-22 17:29:53

Last modified on

2013-03-22 17:29:53

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46H10

Classification

msc 46L05

Synonym

spectral invariance theorem

Synonym

invariance of the spectrum of

C

*

-subalgebras

Defines

invertibility in

C

*

-subalgebras

Generated on Fri Feb 9 21:42:07 2018 by

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spectral permanence theorem

spectral permanence theorem

Let

𝒜

be a unital complex

Banach algebra

and

ℬ

⊆

𝒜

a Banach

subalgebra

that contains the

identity

of

𝒜

.

For every element

x

∈

ℬ

it makes sense to speak of the

spectrum

σ

ℬ

⁢

(

x

)

of

x

relative to

ℬ

as well as the spectrum

σ

𝒜

⁢

(

x

)

of

x

relative to

𝒜

.

We provide here three results of increasing sophistication which relate both these spectrums,

σ

ℬ

⁢

(

x

)

and

σ

𝒜

⁢

(

x

)

. Any of the last two is usually refered to as the

spectral permanence theorem

.

-

Let

ℬ

⊆

𝒜

be as above. For every element

x

∈

ℬ

we have

σ

𝒜

⁢

(

x

)

⊆

σ

ℬ

⁢

(

x

)

.

This first result is purely . It is a straightforward consequence of the fact that invertible elements in

ℬ

are also

invertible

in

𝒜

.

The other inclusion,

σ

ℬ

⁢

(

x

)

⊆

σ

𝒜

⁢

(

x

)

, is not necessarily true. It is true, however, if one considers the

boundary

∂

⁡

σ

ℬ

⁢

(

x

)

instead.

Theorem -

Let

ℬ

⊆

𝒜

be as above. For every element

x

∈

ℬ

we have

∂

⁡

σ

ℬ

⁢

(

x

)

⊆

σ

𝒜

⁢

(

x

)

.

Since the spectrum is a non-empty

compact set

in

ℂ

, one can decompose

ℂ

-

σ

𝒜

⁢

(

x

)

into its

connected components

, obtaining an unbounded

component

Ω

∞

together with a sequence of

bounded

components

Ω

1

,

Ω

2

,

…

,

ℂ

-

σ

𝒜

⁢

(

x

)

=

Ω

∞

∪

Ω

1

∪

Ω

2

∪

⋯

Of course there may be only a finite number of bounded components or none.

Theorem -

Let

x

∈

ℬ

⊆

𝒜

be as above. Then

σ

ℬ

⁢

(

x

)

is obtained from

σ

𝒜

⁢

(

x

)

by adjoining to it some (possibly none) bounded components of

ℂ

-

σ

𝒜

⁢

(

x

)

.

As an example, if

σ

𝒜

⁢

(

x

)

is the unit circle, then

σ

ℬ

⁢

(

x

)

can only possibly be the unit circle or the

closed unit disk

.

Title

spectral permanence theorem

Canonical name

SpectralPermanenceTheorem

Date of creation

2013-03-22 17:29:50

Last modified on

2013-03-22 17:29:50

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46H10

Classification

msc 46H05

Generated on Fri Feb 9 21:42:09 2018 by

LaTeXML
6j-representation in 3D in grafix

6j-representation in 3D in grafix

0.1

Graphical representation of the topological quantum field (TQFT) state on the tetrahedron

Notes:

The

p

⁢

s

⁢

t

⁢

r

⁢

i

⁢

c

⁢

k

⁢

s

code for the graphics is courtesy of pahio.

Arrows indicate the

orientations

defined for the TQFT

state on tetrahedron

by the

6

⁢

j

-symbols.

Title

6j-representation in 3D in grafix

Canonical name

6jrepresentationIn3DInGrafix

Date of creation

2013-03-22 18:22:38

Last modified on

2013-03-22 18:22:38

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

38

Author

bci1 (20947)

Entry type

Example

Classification

msc 46H35

Classification

msc 81T45

Classification

msc 57R56

Synonym

6j-representation in 3D

Related topic

Tetrahedron

Related topic

StateOnTheTetrahedron

Related topic

RegularTetrahedron3

Related topic

TriangleMidSegmentTheorem

Related topic

Heptahedron

Related topic

GergonneTriangle

Related topic

RegularPolyhedron

Related topic

NormalLine

Defines

tetrahedron graphics

Generated on Fri Feb 9 21:42:17 2018 by

LaTeXML
von Neumann double commutant theorem

von Neumann double commutant theorem

The

von Neumann double commutant theorem

is a remarkable result in the theory of

self-adjoint

algebras of

operators

on

Hilbert spaces

, as it expresses purely topological aspects of these algebras in terms of purely algebraic properties.

Theorem

- von Neumann -

Let

H

be a Hilbert space (

http://planetmath.org/HilbertSpace

) and

B

⁢

(

H

)

its algebra of

bounded operators

. Let

ℳ

be a *-

subalgebra

of

B

⁢

(

H

)

that contains the

identity operator

. The following statements are

equivalent

:

1.

ℳ

=

ℳ

′′

, i.e.

ℳ

equals its

double commutant

.

2.

ℳ

is closed in the

weak operator topology

.

3.

ℳ

is closed in the strong operator topology.

Thus, a purely topological property of a

ℳ

, as being closed for some operator topology, is equivalent to a purely algebraic property, such as being equal to its double commutant.

This result is also known as the

bicommutant theorem

or the

von Neumann density theorem

.

Title

von Neumann double commutant theorem

Canonical name

VonNeumannDoubleCommutantTheorem

Date of creation

2013-03-22 18:40:27

Last modified on

2013-03-22 18:40:27

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46H35

Classification

msc 46K05

Classification

msc 46L10

Synonym

double commutant theorem

Synonym

bicommutant theorem

Synonym

von Neumann bicommutant theorem

Synonym

von Neumann density theorem

Generated on Fri Feb 9 21:42:23 2018 by

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Wielandt theorem for unital normed algebras

Wielandt theorem for unital normed algebras

Theorem

.

(Wielandt (1949)) Let

A

be a normed unital algebra (with unit

e

). If

x

,

y

∈

A

then

x

⁢

y

-

y

⁢

x

≠

e

.

Proof.

Assume there are

x

,

y

∈

A

such that

x

⁢

y

-

y

⁢

x

=

e

. Then for all

n

∈

ℕ

we have

x

n

⁢

y

-

y

⁢

x

n

=

n

⁢

x

n

-

1

≠

0

We prove this by

induction

over

n

∈

ℕ

. It holds for

n

=

1

by

assumption

. Assume it is valid for

n

∈

ℕ

. Then

x

n

≠

0

and

x

n

+

1

⁢

y

-

y

⁢

x

n

+

1

=

x

n

⁢

(

x

⁢

y

-

y

⁢

x

)

+

(

x

n

⁢

y

-

y

⁢

x

n

)

⁢

x

=

x

n

⁢

e

+

n

⁢

x

n

-

1

⁢

x

=

x

n

⁢

e

+

n

⁢

x

n

=

(

n

+

e

)

⁢

x

n

From this

identity

it follows that

n

⁢

∥

x

n

-

1

∥

=

∥

x

n

⁢

y

-

y

⁢

x

n

∥

≤

2

⁢

∥

x

n

∥

⁢

∥

y

∥

≤

2

⁢

∥

x

n

-

1

∥

⁢

∥

x

∥

⁢

∥

y

∥

It follows that

n

≤

2

⁢

||

x

||

⁢

||

y

||

for all

n

∈

ℕ

which is impossible.
∎

Corollary.

The

identity operator

on a

Hilbert space

ℋ

cannot be expressed as a commutator of two

bounded linear operators

in

ℒ

⁢

(

ℋ

)

.

Remark.

The above can be understood as a version of the uncertainty principle in one

dimension

. Let

H

=

L

2

⁢

(

ℝ

)

. Let

q

:

H

→

H

be

q

⁢

(

f

)

⁢

(

x

)

:=

x

⁢

f

⁢

(

x

)

with

D

⁢

(

q

)

=

{

f

∈

L

2

⁢

(

ℝ

)

:

x

↦

x

⁢

f

⁢

(

x

)

∈

L

2

⁢

(

ℝ

)

}

, the coordinate

operator

and

p

:

H

→

H

,

p

⁢

(

f

)

⁢

(

x

)

=

-

i

⁢

f

′

⁢

(

x

)

the momentum operator with

D

⁢

(

p

)

:=

{

f

∈

L

2

⁢

(

ℝ

)

:

f

⁢

absolutely continuous

,

f

′

∈

L

2

⁢

(

ℝ

)

}

. It follows that

p

⁢

q

-

q

⁢

p

=

-

i

⁢

id

D

⁢

on

⁢

D

=

D

⁢

(

q

)

∩

D

⁢

(

p

)

According to the corollary

D

=

L

2

⁢

(

ℝ

)

can never be the case.

Title

Wielandt theorem for unital normed algebras

Canonical name

WielandtTheoremForUnitalNormedAlgebras

Date of creation

2013-03-22 19:01:22

Last modified on

2013-03-22 19:01:22

Owner

karstenb (16623)

Last modified by

karstenb (16623)

Numerical id

6

Author

karstenb (16623)

Entry type

Theorem

Classification

msc 46H99

Generated on Fri Feb 9 21:42:25 2018 by

LaTeXML
Stone-Weierstrass theorem for locally compact spaces

Stone-Weierstrass theorem for locally compact spaces

The following results generalize the

Stone-Weierstrass theorem

(and its complex version (

http://planetmath.org/StoneWeierstrassTheoremComplexVersion

)) for locally compact spaces. The cost of this

generalization

is that one no longer deals with all

continuous functions

, but only those that

vanish at infinity

.

Real version

Theorem

-

Let

X

be a locally compact space and

C

0

⁢

(

X

,

ℝ

)

the algebra of continuous functions

X

→

ℝ

that vanish at infinity (

http://planetmath.org/ VanishAtInfinity

), endowed with the sup norm

∥

⋅

∥

∞

. Let

𝒜

be a

subalgebra

of

C

0

⁢

(

X

;

ℝ

)

for which the following conditions hold:

1.

∀

x

,

y

∈

X

,

x

≠

y

,

∃

f

∈

𝒜

:

f

(

x

)

≠

f

(

y

)

, i.e.

𝒜

separates points.

2.

For each

x

∈

X

there exists

f

∈

𝒜

such that

f

⁢

(

x

)

≠

0

.

Then

𝒜

is dense in

C

0

⁢

(

X

;

ℝ

)

.

Complex version

Theorem -

Let

X

be a locally compact space and

C

0

⁢

(

X

)

the algebra of continuous functions

X

→

ℂ

that vanish at infinity, endowed with the sup norm

∥

⋅

∥

∞

. Let

𝒜

be a subalgebra of

C

0

⁢

(

X

)

for which the following conditions hold:

1.

∀

x

,

y

∈

X

,

x

≠

y

,

∃

f

∈

𝒜

:

f

(

x

)

≠

f

(

y

)

, i.e.

𝒜

separates points.

2.

For each

x

∈

X

there exists

f

∈

𝒜

such that

f

⁢

(

x

)

≠

0

.

3.

If

f

∈

𝒜

then

f

¯

∈

𝒜

, i.e.

𝒜

is a self-adjoint subalgebra of

C

⁢

(

X

)

.

Then

𝒜

is dense in

C

0

⁢

(

X

)

.

Title

Stone-Weierstrass theorem for locally compact spaces

Canonical name

StoneWeierstrassTheoremForLocallyCompactSpaces

Date of creation

2013-03-22 18:41:09

Last modified on

2013-03-22 18:41:09

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46J10

Generated on Fri Feb 9 21:42:27 2018 by

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Stone-Weierstrass theorem (complex version)

Stone-Weierstrass theorem (complex version)

Theorem

-

Let

X

be a compact space and

C

⁢

(

X

)

the

algebra

of

continuous functions

X

⟶

ℂ

endowed with the sup norm

∥

⋅

∥

∞

. Let

𝒜

be a

subalgebra

of

C

⁢

(

X

)

for which the following conditions hold:

1.

∀

x

,

y

∈

X

,

x

≠

y

,

∃

f

∈

𝒜

:

f

(

x

)

≠

f

(

y

)

, i.e.

𝒜

separates points

2.

1

∈

𝒜

, i.e.

𝒜

contains all

constant functions

3.

If

f

∈

𝒜

then

f

¯

∈

𝒜

, i.e.

𝒜

is a self-adjoint (

http://planetmath.org/InvolutaryRing

) subalgebra of

C

⁢

(

X

)

Then

𝒜

is dense in

C

⁢

(

X

)

.

Proof:

The proof follows easily from the real version of this theorem (see the parent entry (

http://planetmath.org/StoneWeierstrassTheorem

)).

Let

ℛ

be the set of the real parts of

elements

f

∈

𝒜

, i.e.

ℛ

:=

{

Re

⁢

(

f

)

:

f

∈

𝒜

}

It is clear that

ℛ

contains (it is in fact equal) to the set of the imaginary parts of elements of

𝒜

. This can be seen just by multiplying any

function

f

∈

𝒜

by

-

i

.

We can see that

ℛ

⊆

𝒜

. In fact,

Re

⁢

(

f

)

=

f

+

f

¯

2

and by condition 3 this element belongs to

𝒜

.

Moreover,

ℛ

is a subalgebra of

𝒜

. In fact, since

𝒜

is an algebra, the

product

of two elements

Re

⁢

(

f

)

,

Re

⁢

(

g

)

of

ℛ

gives an element of

𝒜

. But since

Re

⁢

(

f

)

.

Re

⁢

(

g

)

is a real valued function, it must belong to

ℛ

. The same can be said about sums and products by real scalars.

Let us now see that

ℛ

separates points. Since

𝒜

separates points, for every

x

≠

y

in

X

there is a function

f

∈

𝒜

such that

f

⁢

(

x

)

≠

f

⁢

(

y

)

. But this implies that

Re

⁢

(

f

⁢

(

x

)

)

≠

Re

⁢

(

f

⁢

(

y

)

)

or

Im

⁢

(

f

⁢

(

x

)

)

≠

Im

⁢

(

f

⁢

(

y

)

)

, hence there is a function in

ℛ

that separates

x

and

y

.

Of course,

ℛ

contains the constant function

1

.

Hence, we can apply the real version of the

Stone-Weierstrass theorem

to conclude that every real valued function in

X

can be uniformly approximated by elements of

ℛ

.

Let us now see that

𝒜

is dense in

C

⁢

(

X

)

. Let

f

∈

C

⁢

(

X

)

. By the previous observation, both

Re

⁢

(

f

)

and

Im

⁢

(

f

)

are the uniform limits of sequences

{

g

n

}

and

{

h

n

}

in

ℛ

. Hence,

∥

f

-

(

g

n

+

i

⁢

h

n

)

∥

∞

≤

∥

Re

⁢

(

f

)

-

g

n

∥

∞

+

∥

Im

⁢

(

f

)

-

h

n

∥

∞

⟶

0

Of course, the

sequence

{

g

n

+

i

⁢

h

n

}

is in

𝒜

. Hence,

𝒜

is dense in

C

⁢

(

X

)

.

□

Title

Stone-Weierstrass theorem (complex version)

Canonical name

StoneWeierstrassTheoremcomplexVersion

Date of creation

2013-03-22 18:02:31

Last modified on

2013-03-22 18:02:31

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46J10

Generated on Fri Feb 9 21:42:30 2018 by

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Wiener algebra

Wiener algebra

0.0.1

Definition and classification of the Wiener algebra

Let

W

be the space of all complex functions on

[

0

,

2

π

[

whose Fourier series

converges
absolutely

, that is, all

functions

f

:

[

0

,

2

π

[

⟶

ℂ

whose Fourier series

f

⁢

(

t

)

=

∑

n

=

-

∞

+

∞

f

^

⁢

(

n

)

⁢

e

i

⁢

n

⁢

t

is such that

∑

n

|

f

^

⁢

(

n

)

|

<

∞

.

Under

pointwise operations

and the norm

∥

f

∥

=

∑

n

|

f

^

⁢

(

n

)

|

,

W

is a commutative Banach
algebra of

continuous functions

, with an

identity element

.

W

is usually called the

Wiener algebra

.

Theorem

-

W

is

isometrically isomorphic

to the

Banach algebra

ℓ

1

⁢

(

ℤ

)

with the

convolution product

. The

isomorphism

is given by:

(

a

k

)

k

∈

ℤ

⟷

f

⁢

(

t

)

=

∑

n

=

-

∞

+

∞

a

k

⁢

e

i

⁢

n

⁢

t

0.0.2

Wiener’s Theorem

Theorem (Wiener) -

If

f

∈

W

has no zeros then

1

/

f

∈

W

, that is,

1

/

f

has an
absolutely convergent Fourier series.

Proof :

We want to prove that

f

is

invertible

in

W

. As

W

is

commutative

, that is the same as proving
that

f

does not belong to any

maximal ideal

of

W

. Therefore we only need to show that

f

is not in the
kernel of any

multiplicative linear functional

of

W

.

Let

ϕ

be a multiplicative linear functional in

W

. We have that

ϕ

⁢

(

f

)

=

ϕ

⁢

(

∑

n

=

-

∞

+

∞

f

^

⁢

(

n

)

⁢

e

i

⁢

n

⁢

t

)

=

∑

n

=

-

∞

+

∞

f

^

⁢

(

n

)

⁢

ϕ

⁢

(

e

i

⁢

n

⁢

t

)

=

∑

n

=

-

∞

+

∞

f

^

⁢

(

n

)

⁢

ϕ

n

⁢

(

e

i

⁢

t

)

Since

∥

ϕ

∥

=

1

we have that

|

ϕ

⁢

(

e

i

⁢

t

)

|

≤

∥

ϕ

∥

⁢

∥

e

i

⁢

t

∥

=

∥

e

i

⁢

t

∥

=

1

and

|

ϕ

⁢

(

e

-

i

⁢

t

)

|

≤

∥

ϕ

∥

⁢

∥

e

-

i

⁢

t

∥

=

∥

e

-

i

⁢

t

∥

=

1

Since

1

=

|

ϕ

⁢

(

e

i

⁢

t

⁢

e

-

i

⁢

t

)

|

=

|

ϕ

⁢

(

e

i

⁢

t

)

|

⁢

|

ϕ

⁢

(

e

-

i

⁢

t

)

|

we deduce that

|

ϕ

⁢

(

e

i

⁢

t

)

|

=

1

We can conclude that

ϕ

⁢

(

e

i

⁢

t

)

=

e

i

⁢

t

0

for some

t

0

∈

[

0

,

2

π

[

Therefore we obtain

ϕ

⁢

(

f

)

=

∑

n

=

-

∞

+

∞

f

^

⁢

(

n

)

⁢

e

i

⁢

n

⁢

t

0

=

f

⁢

(

t

0

)

which is non-zero by definition of

f

.

We conclude that

f

does not belong to the kernel of any multiplicative linear functional

ϕ

.

□

0.0.3

Remark

The Wiener algebra is a Banach *-

algebra

with the

involution

given by

f

*

⁢

(

t

)

:=

f

⁢

(

-

t

)

¯

, but it is not a

C

*

-algebra (

http://planetmath.org/CAlgebra

) under this involution.

Title

Wiener algebra

Canonical name

WienerAlgebra

Date of creation

2013-03-22 17:22:55

Last modified on

2013-03-22 17:22:55

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

10

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46J10

Classification

msc 43A50

Classification

msc 42A20

Classification

msc 46K05

Defines

Wiener theorem

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