aarjaneiro/planetmath huggingface dataset:
proof of classification of separable Hilbert spaces

proof of classification of separable Hilbert spaces

The strategy will be to show that any

separable

, infinite
dimensional

Hilbert space

H

is

equivalent

to

ℓ

2

, where

ℓ

2

is the space of all square summable

sequences

. Then it
will follow that any two separable, infinite dimensional Hilbert
spaces, being equivalent to the same space, are equivalent to each
other.

Since

H

is separable, there exists a

countable

dense subset

S

of

H

. Choose an enumeration of the elements of

S

as

s

0

,

s

1

,

s

2

,

…

. By the

Gram-Schmidt orthonormalization

procedure, one
can exhibit an

orthonormal set

e

0

,

e

1

,

e

2

,

…

such that
each

e

i

is a finite

linear combination

of the

s

i

’s.

Next, we will demonstrate that Hilbert space spanned by the

e

i

’s
is in fact the whole space

H

. Let

v

be any element of

H

.
Since

S

is dense in

H

, for every integer

n

, there exists an
integer

m

n

such that

∥

v

-

s

m

n

∥

≤

2

-

n

The sequence

(

s

m

0

,

s

m

1

,

s

m

2

,

…

)

is a

Cauchy
sequence

because

∥

s

m

i

-

s

m

j

∥

≤

∥

s

m

i

-

v

∥

+

∥

v

-

s

m

j

∥

≤

2

-

i

+

2

-

j

Hence the limit of this sequence must lie in the Hilbert space spanned by

{

s

0

,

s

1

,

s

2

,

…

}

, which is the same as the Hilbert space
spanned by

{

e

0

,

e

1

,

e

2

,

…

}

. Thus,

{

e

0

,

e

1

,

e

2

,

…

}

is an

orthonormal basis

for

H

.

To any

v

∈

H

associate the sequence

U

⁢

(

v

)

=

(

⟨

v

,

s

0

⟩

,

⟨

v

,

s

1

⟩

,

⟨

v

,

s

2

⟩

,

…

)

.
That this sequence lies in

ℓ

2

follows from the generalized Parseval equality

∥

v

∥

2

=

∑

k

=

0

∞

⟨

v

,

s

k

⟩

which also shows that

∥

U

⁢

(

v

)

∥

ℓ

2

=

∥

v

∥

H

. On the other
hand, let

(

w

0

,

w

1

,

w

2

,

…

)

be an element of

ℓ

2

. Then,
by definition, the sequence of partial sums

(

w

0

2

,

w

0

2

+

w

1

2

,

w

0

2

+

w

1

2

+

w

2

2

,

…

)

is a Cauchy sequence. Since

∥

∑

i

=

0

m

w

i

⁢

e

i

-

∑

i

=

0

n

w

i

⁢

e

i

∥

2

=

∑

i

=

0

m

w

i

2

-

∑

i

=

0

n

w

i

2

if

m

>

n

, the sequence of partial sums of

∑

k

=

0

∞

w

i

⁢

e

i

is also a Cauchy sequence, so

∑

k

=

0

∞

w

i

⁢

e

i

converges

and its limit lies in

H

. Hence the

operator

U

is

invertible

and
is an

isometry

between

H

and

ℓ

2

.

Title

proof of classification of separable Hilbert spaces

Canonical name

ProofOfClassificationOfSeparableHilbertSpaces

Date of creation

2013-03-22 14:34:11

Last modified on

2013-03-22 14:34:11

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 46C15

Related topic

VonNeumannAlgebra

Generated on Fri Feb 9 21:39:08 2018 by

LaTeXML
proof of Hilbert space is uniformly convex space

proof of Hilbert space is uniformly convex space

We prove that in fact an

inner product space

is

uniformly convex

.
Let

ϵ

>

0

,

u

,

v

∈

H

such that

∥

u

∥

≤

1

,

∥

v

∥

≤

1

,

∥

u

-

v

∥

≥

ϵ

. Put

δ

=

1

-

1

2

⁢

4

-

ϵ

2

.
Then

δ

>

0

and by the

parallelogram law

∥

u

+

v

∥

2

=

∥

u

+

v

∥

2

+

∥

u

-

v

∥

2

-

∥

u

-

v

∥

2

=

2

⁢

∥

u

∥

2

+

2

⁢

∥

v

∥

2

-

∥

u

-

v

∥

2

≤

4

-

ϵ

2

=

4

⁢

(

1

-

δ

)

2

.

Hence,

∥

u

+

v

2

∥

≤

1

-

δ

.

Since a

Hilbert space

is an inner product space,
a Hilbert space the conditions of a uniformly convex space.

Title

proof of Hilbert space is uniformly convex space

Canonical name

ProofOfHilbertSpaceIsUniformlyConvexSpace

Date of creation

2013-03-22 15:20:48

Last modified on

2013-03-22 15:20:48

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

16

Author

Mathprof (13753)

Entry type

Proof

Classification

msc 46C15

Classification

msc 46H05

Generated on Fri Feb 9 21:39:10 2018 by

LaTeXML
von Neumann algebra

von Neumann algebra

Definition

Let

H

be an

Hilbert space

, and let

B

⁢

(

H

)

be the *-algebra of

bounded operators

in

H

.

A

von Neumann algebra

(or

W

*

-algebra)

ℳ

is a *-

subalgebra

of

B

⁢

(

H

)

that contains the

identity operator

and

satisfies

one of the following

equivalent

conditions:

1.

ℳ

is closed in the

weak operator topology

.

2.

ℳ

is closed in the strong operator topology.

3.

ℳ

=

ℳ

′′

, i.e.

ℳ

equals its

double commutant

.

The

equivalence

between the above conditions is given by the

von Neumann double commutant theorem

.

Since the weak and strong operator topology are weaker than the norm topology, it follows that every von Neumann algebra is a norm closed *-subalgebra of

B

⁢

(

H

)

. Thus, von Neumann algebras are a particular class of

C

*

-algebras (

http://planetmath.org/CAlgebra

) and the results and tools from the

C

*

theory are also applicable in the setting of von Neumann algebras. Nevertheless, the philosophy behind von Neumann algebras is quite different from that of

C

*

-algebras and the tools and techniques for each theory turn out to be different as well.

Examples:

1.

B

⁢

(

H

)

is itself a von Neumann algebra.

2.

L

∞

⁢

(

ℝ

)

(

http://planetmath.org/LinftyXDmu

) as subalgebra of

B

⁢

(

L

2

⁢

(

ℝ

)

)

is a von Neumann algebra.

Title

von Neumann algebra

Canonical name

VonNeumannAlgebra

Date of creation

2013-03-22 17:21:44

Last modified on

2013-03-22 17:21:44

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

29

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46C15

Classification

msc 46H35

Classification

msc 46L10

Synonym

W

*

-algebra

Related topic

CAlgebra

Related topic

TopologicalAlgebra

Related topic

Commutant

Related topic

GroupoidCDynamicalSystem

Related topic

Algebras2

Related topic

CAlgebra3

Related topic

WeakHopfCAlgebra2

Related topic

HAlgebra

Related topic

LocallyCompactQuantumGroup

Related topic

QuantumGroupsAndVonNeumannAlgebras

Generated on Fri Feb 9 21:39:12 2018 by

LaTeXML
characterization of tight frames in ℝ^n

characterization of tight frames in

ℝ

n

Question 1

.

What conditions must the vectors

{

x

i

}

i

=

1

k

⊂

R

n

satisfy

in order to be a tight frame in

R

n

?

Solution

.

Let

E

be the

k

×

n

matrix whose rows are the vectors

{

x

i

}

i

=

1

k

:

E

=

(

x

i

,

1

⋯

x

i

,

n

⋮

⋱

⋮

x

k

,

1

⋯

x

k

,

n

)

.

Then the tight frame condition

∑

i

=

1

k

|

⟨

x

,

x

i

⟩

|

2

=

A

⁢

∥

x

∥

2

gives

(

E

⁢

x

)

T

⁢

E

⁢

x

=

A

⁢

x

T

⁢

x

for all

x

∈

ℝ

n

, or

E

T

⁢

E

=

A

⁢

I

n

:

E

T

⁢

E

=

(

∑

i

=

1

k

x

i

,

1

⁢

x

i

,

1

⋯

∑

i

=

1

k

x

i

,

1

⁢

x

i

,

k

⋮

⋱

⋮

∑

i

=

1

k

x

i

,

k

⁢

x

i

,

1

⋯

∑

i

=

1

k

x

i

,

k

⁢

x

i

,

k

)

=

(

A

⋯

⁢

0

⁢

⋯

0

⋮

⋱

⋮

0

⋯

⁢

0

⁢

⋯

A

)

=

A

⁢

I

n

.

Therefore, the vectors

{

x

i

=

(

x

i

,

1

⋮

x

i

,

n

)

}

i

=

1

k

⊂

ℝ

n

are an A-tight frame iff the vectors

{

x

i

′

=

(

x

1

,

i

⋮

x

k

,

i

)

}

i

=

1

n

⊂

ℝ

k

,

i.e., the columns of

E

, are all of norm

A

and form an

orthogonal

family.

Title

characterization

of tight frames in

ℝ

n

Canonical

name

CharacterizationOfTightFramesInmathbbRn

Date of creation

2013-03-22 14:27:08

Last modified on

2013-03-22 14:27:08

Owner

swiftset (1337)

Last modified by

swiftset (1337)

Numerical id

5

Author

swiftset (1337)

Entry type

Result

Classification

msc 46C99

Generated on Fri Feb 9 21:39:15 2018 by

LaTeXML
frame

frame

Introduction

The

concept

of a frame is a

generalization

of the concept of an

orthonormal basis

: each vector in the space can be represented as a sum of the

elements

in the frame, but not necessarily uniquely. It is because of this redundancy in representation that frames have found important applications. Surpisingly, despite the fact that frames do not in general consist of

orthonormal vectors

, the frame representation of a vector may still

satisfy

the

Parseval equation

.

Definition

A frame in a

Hilbert space

H

is a

collection

(

x

i

)

i

=

1

n

of vectors in

H

such that there exist

constants

A

,

B

>

0

such that for all

x

∈

H

,

A

⁢

∥

x

∥

2

≤

∑

i

|

⟨

x

,

x

i

⟩

|

2

≤

B

⁢

∥

x

∥

2

The constants

A

,

B

are called the lower and upper

frame bounds

respectively, or the frame constants. The optimal frame constants

A

′

and

B

′

are defined respectively as the

supremum

and

infimum

of all possible lower and upper frame bounds. When

A

=

B

, the frame is a tight frame, or an

A

-tight frame. A

1

-tight frame is referred to as a Parseval frame, and the Parseval equation holds with respect to the frame elements:

x

=

∑

i

=

1

n

⟨

x

,

x

i

⟩

⁢

x

i

for all

x

∈

H

.

Associated Operators (for finite frames)

Let

dim

⁡

H

=

n

and

{

x

i

}

i

=

1

k

be a frame in

H

(

k

≥

n

).

The

analysis

operator

θ

:

H

→

ℂ

k

is the

function

defined such that

θ

:

x

↦

(

⟨

x

,

x

i

⟩

)

i

.

The

synthesis

operator

τ

:

ℂ

k

→

H

is the function defined such that

(

c

i

)

i

k

↦

∑

i

=

1

k

c

i

⁢

x

i

.

τ

=

θ

∗

, that is,

⟨

θ

⁢

x

,

y

⟩

ℂ

k

=

⟨

x

,

τ

⁢

y

⟩

H

for all

x

∈

H

and all

y

∈

ℂ

k

.

⟨

θ

⁢

x

,

y

⟩

ℂ

k

=

⟨

(

⟨

x

,

x

i

⟩

H

)

i

=

1

k

,

(

y

i

)

i

=

1

k

⟩

ℂ

k

=

∑

i

=

1

k

⟨

x

,

x

i

⟩

H

⁢

y

i

¯

=

∑

i

=

1

k

⟨

x

,

y

i

⁢

x

i

⟩

H

=

⟨

x

,

∑

i

=

1

k

y

i

⁢

x

i

⟩

H

=

⟨

x

,

τ

⁢

y

⟩

H

The

frame

operator is defined as the

n

×

n

matrix

θ

∗

⁢

θ

:

H

→

H

such that

θ

∗

⁢

θ

:

x

↦

∑

i

=

1

k

⟨

x

,

x

i

⟩

⁢

x

i

for all

x

∈

H

.

The

Grammian

operator is defined as the composition

θ

⁢

θ

∗

:

ℂ

k

→

ℂ

k

.

The Grammian and frame operators have the same nonzero

eigenvalues

.

Title

frame

Canonical name

Frame1

Date of creation

2013-03-22 14:25:37

Last modified on

2013-03-22 14:25:37

Owner

swiftset (1337)

Last modified by

swiftset (1337)

Numerical id

9

Author

swiftset (1337)

Entry type

Definition

Classification

msc 46C99

Related topic

RieszSequence

Related topic

SetOfSampling

Defines

Parseval frame

Defines

tight frame

Defines

frame constants

Defines

frame bounds

Defines

synthesis operator

Defines

analysis operator

Defines

frame operator

Defines

Grammian

Generated on Fri Feb 9 21:39:17 2018 by

LaTeXML
Gabor frame

Gabor frame

One may be interested in Gabor frames and its related theory if one looks further into the frame

framework

. First, denote a lattice by

Λ

=

A

⁢

ℤ

2

⁢

d

, where

A

is an

invertible matrix

, and let

π

⁢

(

ξ

,

ϕ

)

⁢

f

=

e

2

⁢

π

⁢

i

⁢

ξ

⁢

x

⁢

f

⁢

(

x

-

ϕ

)

Definition

.

Let

g

∈

L

2

⁢

(

R

d

)

be a nonzero window, and let

λ

∈

Λ

, then

G

⁢

(

g

,

λ

)

=

{

π

⁢

(

λ

)

⁢

g

:

λ

∈

Λ

}

is a Gabor system. If

G

⁢

(

g

,

λ

)

is a frame, it’s called a Gabor frame for

L

2

⁢

(

R

d

)

Supose now that one wants to look at a more general framework, and work with functions in

L

2

⁢

(

ℝ

d

,

ℂ

n

)

. Then the

definition

above generalises to

Definition

.

Let

𝐠

∈

L

2

⁢

(

R

d

,

C

n

)

be a nonzero window and let

λ

∈

Λ

, then

𝑮

⁢

(

𝒈

,

λ

)

=

{

π

⁢

(

λ

)

⁢

𝒈

:

λ

∈

Λ

}

is a Gabor super-frame if the frame inequalities hold, where

π

⁢

(

ξ

,

ϕ

)

⁢

𝒈

=

e

2

⁢

π

⁢

ı

⁢

x

⋅

ξ

⁢

(

g

1

⁢

(

x

-

ϕ

)

,

g

2

⁢

(

x

-

ϕ

)

,

…

,

g

n

⁢

(

x

-

ϕ

)

)

and for

𝐟

,

𝐡

∈

L

2

⁢

(

R

d

,

C

n

)

⟨

𝒇

,

𝒉

⟩

L

2

⁢

(

ℝ

d

,

ℂ

n

)

=

∑

i

=

1

n

⟨

f

i

,

h

i

⟩

L

2

⁢

(

ℝ

d

)

References

1

Karlheinz GrÃÂ¶chenig, ”Foundations of Time-Frequency Analysis,”

BirkhhÃÂ¤user

(2000)

Title

Gabor frame

Canonical

name

GaborFrame

Date of creation

2013-03-22 17:08:28

Last modified on

2013-03-22 17:08:28

Owner

ErlendA (6587)

Last modified by

ErlendA (6587)

Numerical id

5

Author

ErlendA (6587)

Entry type

Definition

Classification

msc 46C99

Defines

Gabor frame

Defines

Gabor super-frame

Defines

Vector-valued Gabor frame

Generated on Fri Feb 9 21:39:20 2018 by

LaTeXML
inner product space

inner product space

An

inner product space

(or

pre-Hilbert space

) is a

vector space

(over

ℝ

or

ℂ

)
with an

inner product

⟨

⋅

,

⋅

⟩

.

For example,

ℝ

n

with the familiar

dot product

forms an inner product space.

Every inner product space is also a

normed vector space

,
with the norm defined by

∥

x

∥

:=

⟨

x

,

x

⟩

.
This norm satisfies the

parallelogram law

.

If the metric

∥

x

-

y

∥

induced by the norm is complete (

http://planetmath.org/Complete

),
then the inner product space is called a

Hilbert space

.

The Cauchy–

Schwarz inequality

|

⟨

x

,

y

⟩

|

≤

∥

x

∥

⋅

∥

y

∥

(1)

holds in any inner product space.

According to (1), one can define the

angle between two non-zero vectors

x

and

y

:

cos

⁡

(

x

,

y

)

:=

⟨

x

,

y

⟩

∥

x

∥

⋅

∥

y

∥

.

(2)

This provides that the scalars are the real numbers. In any case, the

perpendiculatity

of the vectors may be defined with the condition

⟨

x

,

y

⟩

=

0

.

Title

inner product space

Canonical name

InnerProductSpace

Date of creation

2013-03-22 12:14:05

Last modified on

2013-03-22 12:14:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

23

Author

CWoo (3771)

Entry type

Definition

Classification

msc 46C99

Synonym

pre-Hilbert space

Related topic

InnerProduct

Related topic

OrthonormalBasis

Related topic

HilbertSpace

Related topic

EuclideanVectorSpace2

Related topic

AngleBetweenTwoLines

Related topic

FluxOfVectorField

Related topic

CauchySchwarzInequality

Defines

angle between two vectors

Defines

perpendicularity

Generated on Fri Feb 9 21:39:22 2018 by

LaTeXML
multiresolution analysis

multiresolution analysis

Definition

A

multiresolution analysis

is a

sequence

(

V

j

)

j

∈

ℤ

of

subspaces

of

L

2

⁢

(

ℝ

)

such that

1.

(nesting)

…

⊂

V

-

1

⊂

V

0

⊂

V

1

⊂

…

2.

(

density

)

span

⋃

j

∈

ℤ

V

j

¯

=

L

2

⁢

(

ℝ

)

3.

(

separation

)

⋂

j

∈

ℤ

V

j

=

{

0

}

4.

(

scaling

)

f

⁢

(

x

)

∈

V

j

if and only if

f

⁢

(

2

-

j

⁢

x

)

∈

V

0

5.

(

orthonormal basis

) there exists a

function

Φ

∈

V

0

, called a

scaling function

, such that the system

{

Φ

(

t

-

m

)

}

m

∈

ℤ

}

is an orthonormal basis in

V

0

.

Notes

Multiresolution analysis, particularly scaling functions, are used to derive

wavelets

. The

V

j

are called approximation spaces. Several choices of scaling functions may exist for a given set of approximation spaces— each determines a unique multiresolution analysis.

Title

multiresolution analysis

Canonical name

MultiresolutionAnalysis

Date of creation

2013-03-22 14:26:48

Last modified on

2013-03-22 14:26:48

Owner

swiftset (1337)

Last modified by

swiftset (1337)

Numerical id

5

Author

swiftset (1337)

Entry type

Definition

Classification

msc 46C99

Synonym

level of detail

Related topic

Wavelet

Defines

scaling function

Generated on Fri Feb 9 21:39:24 2018 by

LaTeXML
proof of Riesz representation theorem

proof of Riesz representation theorem

Existence -

If

f

=

0

we can just take

u

=

0

and thereby have

f

⁢

(

x

)

=

0

=

⟨

x

,

0

⟩

for all

x

∈

ℋ

.

Suppose now

f

≠

0

, i.e.

K

⁢

e

⁢

r

⁢

f

≠

ℋ

.

Recall that, since

f

is

continuous

(

http://planetmath.org/ContinuousMap

),

K

⁢

e

⁢

r

⁢

f

is a closed subspace of

ℋ

(

continuity

of

f

implies that

f

-

1

⁢

(

0

)

is closed in

ℋ

). It then follows from the

orthogonal decomposition theorem

that

ℋ

=

K

⁢

e

⁢

r

⁢

f

⊕

(

K

⁢

e

⁢

r

⁢

f

)

⟂

and as

K

⁢

e

⁢

r

⁢

f

≠

ℋ

we can find

z

∈

(

K

⁢

e

⁢

r

⁢

f

)

⟂

such that

∥

z

∥

=

1

.

It follows easily from the linearity of

f

that for every

x

∈

ℋ

we have

f

⁢

(

x

)

⁢

z

-

f

⁢

(

z

)

⁢

x

∈

K

⁢

e

⁢

r

⁢

f

and since

z

∈

(

K

⁢

e

⁢

r

⁢

f

)

⟂

0

=

⟨

f

⁢

(

x

)

⁢

z

-

f

⁢

(

z

)

⁢

x

,

z

⟩

=

f

⁢

(

x

)

⁢

⟨

z

,

z

⟩

-

f

⁢

(

z

)

⁢

⟨

x

,

z

⟩

=

f

⁢

(

x

)

⁢

∥

z

∥

2

-

⟨

x

,

f

⁢

(

z

)

¯

⁢

z

⟩

=

f

⁢

(

x

)

-

⟨

x

,

f

⁢

(

z

)

¯

⁢

z

⟩

which implies

f

⁢

(

x

)

=

⟨

x

,

f

⁢

(

z

)

¯

⁢

z

⟩

.

The theorem then follows by taking

u

=

f

⁢

(

z

)

¯

⁢

z

.

Uniqueness -

Suppose there were

u

1

,

u

2

∈

ℋ

such that for every

x

∈

ℋ

f

⁢

(

x

)

=

⟨

x

,

u

1

⟩

=

⟨

x

,

u

2

⟩

.

Then

⟨

x

,

u

1

-

u

2

⟩

=

0

for every

x

∈

ℋ

. Taking

x

=

u

1

-

u

2

we obtain

∥

u

1

-

u

2

∥

2

=

0

, which implies

u

1

=

u

2

.

□

Title

proof of Riesz representation theorem

Canonical name

ProofOfRieszRepresentationTheorem

Date of creation

2013-03-22 17:32:37

Last modified on

2013-03-22 17:32:37

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Proof

Classification

msc 46C99

Generated on Fri Feb 9 21:39:26 2018 by

LaTeXML
proof of Riesz representation theorem for separable Hilbert spaces

proof of Riesz representation theorem for separable Hilbert spaces

Let

{

𝐞

0

,

𝐞

1

,

𝐞

2

,

…

}

be an

orthonormal basis

for the

Hilbert space

ℋ

. Define

c

i

=

f

⁢

(

𝐞

i

)

and

v

=

∑

k

=

0

n

c

¯

i

⁢

𝐞

i

.

The

linear map

(

http://planetmath.org/ContinuousLinearMapping

)

f

is

continuous

if and only if it is

bounded

, i.e. there exists a constant

C

such that

|

f

⁢

(

v

)

|

≤

C

⁢

∥

v

∥

. Then

f

⁢

(

v

)

=

∑

k

=

0

n

c

¯

k

⁢

f

⁢

(

𝐞

k

)

=

∑

k

=

0

n

|

c

k

|

2

≤

C

⁢

∑

k

=

0

n

|

c

k

|

2

.

Simplifying,

∑

k

=

0

n

|

c

k

|

2

≤

C

2

. Hence

∑

k

=

0

∞

c

k

⁢

𝐞

k

converges to an element

u

in

H

.

For every basis element,

f

⁢

(

𝐞

i

)

=

c

k

=

⟨

u

,

𝐞

i

⟩

. By linearity, it will also be true that

f

⁢

(

v

)

=

⟨

u

,

v

⟩

⁢

if

v

is a finite superposition of basis vectors.

Any vector in the Hilbert space can be written as the limit of a sequence of finite superpositions of basis vectors hence, by

continuity

,

f

⁢

(

v

)

=

⟨

u

,

v

⟩

⁢

for all

⁢

v

∈

ℋ

It is easy to see that

u

is unique. Suppose there existed two vectors

u

1

and

u

2

such that

f

⁢

(

v

)

=

⟨

u

1

,

v

⟩

=

⟨

u

2

,

v

⟩

. Then

⟨

u

1

-

u

2

,

v

⟩

=

0

for all vectors

v

∈

ℋ

. But then,

⟨

u

1

-

u

2

,

u

1

-

u

2

⟩

=

0

which is only possible if

u

1

-

u

2

=

0

, i.e. if

u

1

=

u

2

.

Title

proof of Riesz representation theorem for separable Hilbert spaces

Canonical name

ProofOfRieszRepresentationTheoremForSeparableHilbertSpaces

Date of creation

2013-03-22 14:34:20

Last modified on

2013-03-22 14:34:20

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Proof

Classification

msc 46C99

Generated on Fri Feb 9 21:39:29 2018 by

LaTeXML
Riesz-Fischer theorem

Riesz-Fischer theorem

Let

{

e

n

}

be an

orthonormal basis

for a (real or complex)

infinite-dimensional

Hilbert space

ℋ

. If

{

c

n

}

is a sequence of (real or complex) numbers such that

∑

|

c

n

|

2

converges, then there is an

x

∈

ℋ

such that

x

=

∑

n

=

1

∞

c

n

⁢

e

n

, and

c

n

=

⟨

x

,

e

n

⟩

.

Title

Riesz-Fischer theorem

Canonical name

RieszFischerTheorem

Date of creation

2013-03-22 14:09:46

Last modified on

2013-03-22 14:09:46

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

7

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 46C99

Related topic

LpSpace

Related topic

L2SpacesAreHilbertSpaces

Related topic

HilbertSpace

Related topic

EllpXSpace

Related topic

ClassificationOfHilbertSpaces

Generated on Fri Feb 9 21:39:31 2018 by

LaTeXML
Riesz representation theorem

Riesz representation theorem

For every

continuous

linear functional

f

on a

Hilbert space

ℋ

, there is a unique

u

∈

ℋ

such that

f

⁢

(

x

)

=

⟨

x

,

u

⟩

for all

x

∈

ℋ

.

Note:

⟨

x

,

u

⟩

denotes the

inner product

between

x

and

u

.

Title

Riesz representation theorem

Canonical name

RieszRepresentationTheorem

Date of creation

2013-03-22 14:09:43

Last modified on

2013-03-22 14:09:43

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

9

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 46C99

Related topic

RieszFischerTheorem

Generated on Fri Feb 9 21:39:32 2018 by

LaTeXML
set of sampling

set of sampling

Definition

Let

F

be a

Hilbert space

of

functions

defined on a domain

D

. Let

T

=

{

t

i

}

i

∈

I

be a finite or

infinite

sequence

of points in

D

.

T

is said to be a

set of sampling

for

F

if the sampling

operator

S

:

F

→

l

|

T

|

2

defined by

S

:

f

↦

(

f

⁢

(

t

1

)

f

⁢

(

t

2

)

⋮

)

is

bounded

(i.e.

continuous

) and bounded below; i.e. if

∃

A

,

B

>

0

⁢

such that

⁢

∀

f

∈

F

,

A

⁢

∥

f

∥

2

≤

∑

i

=

1

|

T

|

|

f

⁢

(

t

i

)

|

2

≤

B

⁢

∥

f

∥

2

.

Relation to Frames

Using the

Riesz Representation Theorem

, it is easy to show that every set of sampling determines a unique frame in such a way that the

analysis operator

of that frame is the sampling operator associated with the set of sampling. In fact, let

t

=

{

t

i

}

be a set of sampling with sampling operator

S

t

. Use the Riesz representation theorem to rewrite

S

t

in terms of vectors

{

g

i

}

in

F

:

S

:

f

↦

(

f

⁢

(

t

1

)

f

⁢

(

t

2

)

⋮

)

=

(

⟨

f

,

g

1

⟩

⟨

f

,

g

2

⟩

⋮

)

then note that

∀

f

∈

F

,

A

⁢

∥

f

∥

2

≤

∑

i

|

⟨

f

,

g

i

⟩

|

2

≤

B

⁢

∥

f

∥

2

,

so the

{

g

i

}

form a frame with bounds

A

,

B

, and

S

t

=

θ

g

.

Reconstruction

Particularly nice sets of sampling are those that correspond to tight frames, because then

θ

g

∗

⁢

θ

g

=

θ

g

∗

⁢

S

t

=

A

⁢

I

, and it is possible to reconstruct the function

f

, given its values over the set of sampling:

f

=

1

A

⁢

∑

i

f

⁢

(

t

i

)

⁢

g

i

.

Sets of sampling which correspond to tight frames are referred to as tight sets of sampling.

Title

set of sampling

Canonical name

SetOfSampling

Date of creation

2013-03-22 14:27:50

Last modified on

2013-03-22 14:27:50

Owner

swiftset (1337)

Last modified by

swiftset (1337)

Numerical id

4

Author

swiftset (1337)

Entry type

Definition

Classification

msc 46C99

Synonym

sampling set

Related topic

Frame2

Defines

set of sampling

Defines

sampling operator

Generated on Fri Feb 9 21:39:35 2018 by

LaTeXML