aarjaneiro/planetmath huggingface dataset:
pointwise limit of bounded operators is bounded

pointwise limit of bounded operators is bounded

The following result is a corollary of the

principle of uniform boundedness

.

Theorem -

Let

X

be a

Banach space

and

Y

a

normed vector space

. Let

(

T

n

)

∈

B

⁢

(

X

,

Y

)

be a

sequence

of

bounded operators

from

X

to

Y

. If

(

T

n

⁢

x

)

converges for every

x

∈

X

, then the

operator

T

:

X

⟶

Y

T

⁢

x

=

lim

n

→

∞

⁡

T

n

⁢

x

is linear and . Moreover, the sequence

(

∥

T

n

∥

)

is

bounded

(

http://planetmath.org/Bounded

).

Proof :

It is clear that the operator

T

is linear.

For each

x

∈

X

we have

sup

n

⁡

∥

T

n

⁢

x

∥

<

∞

since

(

T

n

⁢

x

)

is . By the principle of uniform boundedness (

http://planetmath.org/BanachSteinhausTheorem

) we must also have

M

:=

sup

n

⁡

∥

T

n

∥

<

∞

.

Then for each

x

∈

X

we have

∥

T

⁢

x

∥

=

lim

n

→

∞

⁡

∥

T

n

⁢

x

∥

≤

M

⁢

∥

x

∥

which means that

T

is .

□

Title

pointwise limit of bounded operators is bounded

Canonical name

PointwiseLimitOfBoundedOperatorsIsBounded

Date of creation

2013-03-22 17:32:10

Last modified on

2013-03-22 17:32:10

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Corollary

Classification

msc 46B99

Classification

msc 47A05

Generated on Fri Feb 9 21:37:42 2018 by

LaTeXML
proof of Banach-Steinhaus theorem

proof of Banach-Steinhaus theorem

Let

E

n

=

{

x

∈

X

:

∥

T

⁢

(

x

)

∥

≤

n

⁢

for all

⁢

T

∈

ℱ

}

.

From the

hypothesis

, we have that

⋃

n

=

1

∞

E

n

=

X

.

Also, each

E

n

is closed, since it can be written as

E

n

=

⋂

T

∈

ℱ

T

-

1

⁢

(

B

⁢

(

0

,

n

)

)

,

where

B

⁢

(

0

,

n

)

is the closed ball centered at

0

with radius

n

in

Y

,
and each of the sets in the

intersection

is closed due to the continuity of the

operators

.
Now since

X

is a

Banach space

, Baire’s category

theorem

implies that there exists

n

such that

E

n

has
nonempty interior. So there is

x

0

∈

E

n

and

r

>

0

such
that

B

⁢

(

x

0

,

r

)

⊂

E

n

. Thus if

∥

x

∥

≤

r

, we have

∥

T

⁢

(

x

)

∥

-

∥

T

⁢

(

x

0

)

∥

≤

∥

T

⁢

(

x

0

)

+

T

⁢

(

x

)

∥

=

∥

T

⁢

(

x

0

+

x

)

∥

≤

n

for each

T

∈

ℱ

, and so

∥

T

⁢

(

x

)

∥

≤

n

+

∥

T

⁢

(

x

0

)

∥

so if

∥

x

∥

≤

1

, we have

∥

T

⁢

(

x

)

∥

=

1

r

⁢

∥

T

⁢

(

r

⁢

x

)

∥

≤

1

r

⁢

(

n

+

∥

T

⁢

(

x

0

)

∥

)

=

c

,

and this means that

∥

T

∥

=

sup

⁡

{

∥

T

⁢

x

∥

:

∥

x

∥

≤

1

}

≤

c

for all

T

∈

ℱ

.

Title

proof of Banach-Steinhaus theorem

Canonical

name

ProofOfBanachSteinhausTheorem

Date of creation

2013-03-22 14:48:41

Last modified on

2013-03-22 14:48:41

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Proof

Classification

msc 46B99

Generated on Fri Feb 9 21:37:44 2018 by

LaTeXML
proof of necessary and sufficient conditions for a normed vector space to be a Banach space

proof of necessary and sufficient conditions for a normed vector space to be a Banach space

We prove here that in order for a

normed space

, say

X

, with the norm, say

∥

⋅

∥

to be
Banach, it is necessary and sufficient that

convergence

of every absolutely convergent series in

X

implies convergence of the series in

X

.

Suppose that

X

is Banach. Let a

sequence

(

x

n

)

be in

X

such that the series

∑

n

∥

x

n

∥

converges

. Then for all

ϵ

>

0

there exists

N

such that for all

m

>

n

>

N

we have

∥

∑

n

+

1

m

x

n

∥

≤

∑

n

+

1

m

∥

x

n

∥

<

ϵ

Hence

s

k

=

∑

n

=

1

k

x

n

is a

Cauchy sequence

in

X

. Since

X

is Banach,

s

k

converges in

X

.

Conversely, suppose that

absolute convergence

implies convergence. Let

(

x

n

)

be a Cauchy sequence
in

X

. Then for all

m

≥

1

there exists

N

m

such that for all

k

,

k

′

≥

N

m

we have

∥

x

k

-

x

k

′

∥

<

1

/

m

2

.
We’ll conveniently choose

N

m

so that

N

m

is an

increasing sequence

in

m

. Then in particular,

∥

x

N

m

-

x

N

m

+

1

∥

<

1

/

m

2

. Hence we have,

∑

m

=

1

M

∥

x

N

m

-

x

N

m

+

1

∥

<

∑

m

=

1

M

1

m

2

The sum on the right converges, so must the sum on the left. Since absolute convergence implies convergence,
we must have

∑

m

=

1

M

(

x

N

m

-

x

N

m

+

1

)

converges as

M

tends to infinity. So there is an

s

in

X

which is the limit of the sum above. As a telescoping
series, however, the sum above converges to

lim

M

→

∞

⁡

(

x

N

1

-

x

N

m

)

=

s

. Since

s

and

x

N

1

are both in

X

, so is the limit of

x

N

m

, which is a

subsequence

of the Cauchy sequence

(

x

n

)

. Hence

(

x

n

)

converges in

X

. So

X

is Banach.

This

completes

the proof.

Title

proof of necessary and sufficient conditions for a

normed vector space

to be a

Banach space

Canonical name

ProofOfNecessaryAndSufficientConditionsForANormedVectorSpaceToBeABanachSpace

Date of creation

2013-03-22 17:35:11

Last modified on

2013-03-22 17:35:11

Owner

willny (13192)

Last modified by

willny (13192)

Numerical id

4

Author

willny (13192)

Entry type

Proof

Classification

msc 46B99

Generated on Fri Feb 9 21:37:46 2018 by

LaTeXML
quotient norm

quotient norm

Let

V

be a

normed vector space

with norm

∥

⋅

∥

. Let

M

be a closed subspace of

V

and

V

/

M

the

quotient vector space

.

The norm

∥

⋅

∥

induces a norm

∥

⋅

∥

V

/

M

in

V

/

M

, called the

quotient norm

, given by

∥

v

+

M

∥

V

/

M

:=

inf

u

∈

v

+

M

⁡

∥

u

∥

=

inf

m

∈

M

⁡

∥

v

+

m

∥

Theorem -

∥

⋅

∥

V

/

M

is a norm in

V

/

M

iff

M

is closed in

V

.

Title

quotient norm

Canonical name

QuotientNorm

Date of creation

2013-03-22 17:22:58

Last modified on

2013-03-22 17:22:58

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46B99

Generated on Fri Feb 9 21:37:48 2018 by

LaTeXML
quotients of Banach spaces by closed subspaces are Banach spaces under the quotient norm

quotients of Banach spaces by closed subspaces are Banach spaces under the quotient norm

Theorem

-

Let

X

be a

Banach space

and

M

a closed subspace. Then

X

/

M

with the

quotient norm

is a
Banach space.

Proof :

In to prove that

X

/

M

is a Banach space it is enough to prove that every series in

X

/

M

that

converges absolutely

also

converges

in

X

/

M

.

Let

∑

n

X

n

be an absolutely convergent series in

X

/

M

, i.e.,

∑

n

∥

X

n

∥

X

/

M

<

∞

.
By

definition

of the quotient norm, there exists

x

n

∈

X

n

such that

∥

x

n

∥

≤

∥

X

n

∥

X

/

M

+

2

-

n

It is clear that

∑

n

∥

x

n

∥

<

∞

and so, as

X

is a Banach space,

∑

n

x

n

is
convergent.

Let

x

=

∑

n

x

n

and

s

k

=

∑

n

=

1

k

x

n

. We have that

x

-

s

k

+

M

=

(

x

+

M

)

-

(

s

k

+

M

)

=

(

x

+

M

)

-

∑

n

=

1

k

(

x

n

+

M

)

=

(

x

+

M

)

-

∑

n

=

1

k

X

n

Since

∥

x

-

s

k

+

M

∥

X

/

M

≤

∥

x

-

s

k

∥

⟶

0

we see that

∑

n

X

n

converges in

X

/

M

to

x

+

M

.

□

Title

quotients

of Banach spaces by closed subspaces are Banach spaces under the quotient norm

Canonical

name

QuotientsOfBanachSpacesByClosedSubspacesAreBanachSpacesUnderTheQuotientNorm

Date of creation

2013-03-22 17:23:01

Last modified on

2013-03-22 17:23:01

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46B99

Generated on Fri Feb 9 21:37:50 2018 by

LaTeXML
scaling of the open ball in a normed vector space

scaling of the open ball in a normed vector space

Let

V

be a

vector space

over a field

F

(real or complex), and let

∥

⋅

∥

be a norm on

V

. Further,
for

r

>

0

,

v

∈

V

, let

B

r

⁢

(

v

)

=

{

w

∈

V

:

∥

w

-

v

∥

<

r

}

.

Then for any non-zero

λ

∈

F

, we have

λ

⁢

B

r

⁢

(

v

)

=

B

|

λ

|

⁢

r

⁢

(

λ

⁢

v

)

.

The claim is clear for

λ

=

0

, so we can assume that

λ

≠

0

.
Then

λ

⁢

B

r

⁢

(

v

)

=

{

z

∈

V

:

∥

w

-

v

∥

<

r

and

z

=

λ

w

}

=

{

z

∈

V

:

∥

z

λ

-

v

∥

<

r

}

=

{

z

∈

V

:

∥

z

-

λ

v

∥

<

|

λ

|

r

}

=

B

|

λ

|

⁢

r

⁢

(

λ

⁢

v

)

.

Title

scaling of the open ball in a

normed vector space

Canonical name

ScalingOfTheOpenBallInANormedVectorSpace

Date of creation

2013-03-22 15:33:25

Last modified on

2013-03-22 15:33:25

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Theorem

Classification

msc 46B99

Generated on Fri Feb 9 21:37:53 2018 by

LaTeXML
shift operators in ℓ^p

shift operators in

ℓ

p

Let

𝔽

be

ℝ

or

ℂ

, and let

1

≤

p

≤

∞

,
let

ℓ

p

(

𝔽

)

,

∥

⋅

∥

p

be as in the parent entry.

The right and left

shift operators

S

r

,

S

l

:

ℓ

p

⁢

(

𝔽

)

→

ℓ

p

⁢

(

𝔽

)

as defined as follows. For

a

=

(

a

1

,

a

2

,

…

)

∈

ℓ

p

⁢

(

𝔽

)

,

S

r

⁢

(

a

)

=

(

0

,

a

1

,

a

2

,

…

)

and

S

l

⁢

(

a

)

=

(

a

2

,

a

3

,

…

)

.

Properties

1.

S

l

∘

S

r

is the identity, but

S

r

∘

S

l

is not.

2.

S

r

is an isometry;

∥

S

r

⁢

(

a

)

∥

=

∥

a

∥

,
and

∥

S

l

⁢

(

a

)

∥

p

≤

∥

a

∥

. Both shift operators
are therefore bounded (and

continuous

).

Title

shift operators in

ℓ

p

Canonical name

ShiftOperatorsInellp

Date of creation

2013-03-22 15:17:40

Last modified on

2013-03-22 15:17:40

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

4

Author

matte (1858)

Entry type

Definition

Classification

msc 46B99

Classification

msc 54E50

Generated on Fri Feb 9 21:37:55 2018 by

LaTeXML
all orthonormal bases have the same cardinality

all orthonormal bases have the same cardinality

Theorem

. –

All orthonormal bases of an

Hilbert space

H

have the same

cardinality

. It follows that the

concept

of

dimension

of a Hilbert space is

well-defined

.

Proof:

When

H

is finite-dimensional (as a

vector space

), every

orthonormal basis

is a

Hamel basis

of

H

. Thus, the result follows from the fact that all Hamel bases of a vector space have the same cardinality (see this entry (

http://planetmath.org/AllBasesForAVectorSpaceHaveTheSameCardinality

)).

We now consider the case where

H

is infinite-dimensional (as a vector space). Let

{

e

i

}

i

∈

I

and

{

f

j

}

j

∈

J

be two orthonormal basis of

H

,

indexed by

the sets

I

and

J

, respectively. Since

H

is infinite dimensional the sets

I

and

J

must be

infinite

.

We know, from Parseval’s equality, that for every

x

∈

H

∥

x

∥

2

=

∑

i

∈

I

|

⟨

x

,

e

i

⟩

|

2

We know that, in the above sum,

⟨

x

,

e

i

⟩

≠

0

for only a

countable

number of

i

∈

I

. Thus, considering

x

as

f

j

, the set

I

j

:=

{

i

∈

I

:

⟨

f

j

,

e

i

⟩

≠

0

}

is countable. Since for each

i

∈

I

we also have

∥

e

i

∥

2

=

∑

j

∈

J

|

⟨

e

i

,

f

j

⟩

|

2

there must be

j

∈

J

such that

⟨

f

j

,

e

i

⟩

≠

0

. We conclude that

I

=

⋃

j

∈

J

I

j

.

Hence, since each

I

j

is countable,

I

≤

J

×

ℕ

≅

J

(because

J

is infinite).

An analogous proves that

J

≤

I

. Hence, by the

Schroeder-Bernstein theorem

J

and

I

have the same cardinality.

□

Title

all orthonormal bases have the same cardinality

Canonical

name

AllOrthonormalBasesHaveTheSameCardinality

Date of creation

2013-03-22 17:56:10

Last modified on

2013-03-22 17:56:10

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46C05

Synonym

dimension of an Hilbert space is well-defined

Generated on Fri Feb 9 21:37:57 2018 by

LaTeXML
Bessel inequality

Bessel inequality

Let

ℋ

be a

Hilbert space

, and suppose

e

1

,

e

2

,

…

∈

ℋ

is an orthonormal sequence. Then for any

x

∈

ℋ

,

∑

k

=

1

∞

|

⟨

x

,

e

k

⟩

|

2

≤

∥

x

∥

2

.

Bessel’s

inequality

immediately lets us define the sum

x

′

=

∑

k

=

1

∞

⟨

x

,

e

k

⟩

⁢

e

k

.

The inequality means that the series converges.

For a

complete

orthonormal series, we have Parseval’s theorem, which replaces inequality with equality (and consequently

x

′

with

x

).

Title

Bessel inequality

Canonical name

BesselInequality

Date of creation

2013-03-22 12:46:38

Last modified on

2013-03-22 12:46:38

Owner

ariels (338)

Last modified by

ariels (338)

Numerical id

5

Author

ariels (338)

Entry type

Theorem

Classification

msc 46C05

Generated on Fri Feb 9 21:37:59 2018 by

LaTeXML
direct sum of bounded operators on Hilbert spaces

direct sum of bounded operators on Hilbert spaces

\PMlinkescapephrase

direct sum

0.1

Definition

Let

{

H

i

}

i

∈

I

be a family of

Hilbert spaces

indexed by

a set

I

. For each

i

∈

I

let

T

i

:

H

i

⟶

H

i

be a

bounded linear operator

on

H

i

such that the family

{

T

i

}

i

∈

I

of bounded linear operators is uniformly bounded, i.e.

sup

⁡

{

∥

T

i

∥

:

i

∈

I

}

<

∞

.

Definition -

The

direct sum

of the uniformly bounded family

{

T

i

}

i

∈

I

is the

operator

⊕

i

∈

I

T

i

:

⊕

i

∈

I

H

i

⟶

⊕

i

∈

I

H

i

on the

direct sum of Hilbert spaces

⊕

i

∈

I

H

i

defined by

(

⊕

i

∈

I

T

i

⁢

(

x

)

)

i

:=

T

i

⁢

x

i

It can be seen that

⊕

i

∈

I

T

i

is well-defined and is in fact a bounded linear operator, whose norm is

∥

⊕

i

∈

I

T

i

∥

=

sup

⁡

{

∥

T

i

∥

:

i

∈

I

}

0.2

Properties

•

⊕

i

∈

I

(

a

⁢

T

i

+

b

⁢

S

i

)

=

a

⁢

⊕

i

∈

I

T

i

+

b

⁢

⊕

i

∈

I

S

i

, where

a

,

b

∈

ℂ

.

•

(

⊕

i

∈

I

T

i

)

*

=

⊕

i

∈

I

T

i

*

.

•

(

⊕

i

∈

I

T

i

)

⁢

(

⊕

i

∈

I

S

i

)

=

⊕

i

∈

I

T

i

⁢

S

i

.

Title

direct sum of bounded operators on Hilbert spaces

Canonical name

DirectSumOfBoundedOperatorsOnHilbertSpaces

Date of creation

2013-03-22 18:00:32

Last modified on

2013-03-22 18:00:32

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46C05

Classification

msc 47A05

Generated on Fri Feb 9 21:38:07 2018 by

LaTeXML
direct sum of Hilbert spaces

direct sum of Hilbert spaces

Let

{

H

i

}

i

∈

I

be a family of

Hilbert spaces

indexed by

a set

I

. The

direct sum

of this family of Hilbert spaces, denoted as

⊕

i

∈

I

H

i

consists of all

elements

v

of the

Cartesian product

(

http://planetmath.org/GeneralizedCartesianProduct

) of

{

H

i

}

i

∈

I

such that

∑

∥

v

i

∥

2

<

∞

. Of course, for the previous sum to be finite only at most a

countable

number of

v

i

can be non-zero.

Vector addition

and

scalar multiplication

are defined termwise: If

u

,

v

∈

⊕

i

∈

I

H

i

, then

(

u

+

v

)

i

=

u

i

+

v

i

and

(

s

⁢

v

)

i

=

s

⁢

v

i

.

The

inner product

of two vectors is defined as

⟨

u

,

v

⟩

=

∑

i

∈

I

⟨

u

i

,

v

i

⟩

Linked PDF file:

http://images.planetmath.org/cache/objects/6363/pdf/DirectSumOfHilbertSpaces.pdf

Title

direct sum of Hilbert spaces

Canonical name

DirectSumOfHilbertSpaces

Date of creation

2013-03-22 14:43:55

Last modified on

2013-03-22 14:43:55

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

10

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46C05

Related topic

CategoryOfHilbertSpaces

Generated on Fri Feb 9 21:38:09 2018 by

LaTeXML
every Hilbert space has an orthonormal basis

every Hilbert space has an orthonormal basis

Theorem -

Every

Hilbert space

H

≠

{

0

}

has an

orthonormal basis

.

Proof :

As could be expected, the proof makes use of Zorn’s Lemma. Let

𝒪

be the set of all

orthonormal sets

of

H

. It is clear that

𝒪

is non-empty since the set

{

x

}

is in

𝒪

, where

x

is an element of

H

such that

∥

x

∥

=

1

.

The elements of

𝒪

can be ordered by inclusion, and each chain

𝒞

in

𝒪

has an

upper bound

, given by the union of all elements of

𝒞

. Thus, Zorn’s Lemma assures the existence of a

maximal element

B

in

𝒪

. We claim that

B

is an orthonormal basis of

H

.

It is clear that

B

is an orthonormal set, as it belongs to

𝒪

. It remains to see that the

linear span

of

B

is

dense in

H

.

Let

span

⁢

B

¯

denote the

closure

of the span of

B

. Suppose

span

⁢

B

¯

≠

H

. By the

orthogonal decomposition theorem

we know that

H

=

span

⁢

B

¯

⊕

(

span

⁢

B

¯

)

⟂

Thus, we conclude that

(

span

⁢

B

¯

)

⟂

≠

{

0

}

, i.e. there are elements which are

orthogonal

(

http://planetmath.org/OrthogonalVectors

) to

span

⁢

B

¯

. This contradicts the maximality of

B

since, by picking an element

y

∈

(

span

⁢

B

¯

)

⟂

with

∥

y

∥

=

1

,

B

∪

{

y

}

would belong belong to

𝒪

and would be greater than

B

.

Hence,

span

⁢

B

¯

=

H

, and this finishes the proof.

□

Title

every Hilbert space has an orthonormal basis

Canonical name

EveryHilbertSpaceHasAnOrthonormalBasis

Date of creation

2013-03-22 17:56:05

Last modified on

2013-03-22 17:56:05

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46C05

Generated on Fri Feb 9 21:38:10 2018 by

LaTeXML
example of non-separable Hilbert space

example of non-separable Hilbert space

As an exaple of a

Hilbert space

which is not

separable

, one may
consider the following

function space

:

Consider real-valued functions on the real line but, instead of the
usual

L

2

norm, use the following

inner product

:

(

f

,

g

)

=

lim

R

→

∞

⁡

1

R

⁢

∫

-

R

+

R

f

⁢

(

x

)

⁢

g

⁢

(

x

)

⁢

𝑑

x

The first thing to note about this is that non-trivial

functions

have
norm 0. For

instance

, any function of

L

2

has zero norm according
to this inner product.

Define the Hilbert space as the set of

equivalence classes

of
functions for which this norm is finite modulo functions for which it
is zero. Note that

sin

⁡

a

⁢

x

and

sin

⁡

b

⁢

x

are

orthogonal

under this
norm if

a

≠

b

. Hence, the set of functions

sin

⁡

a

⁢

x

, where

a

is a

real number

, form an

orthonormal set

. Since the number of real
numbers is

uncountable

, we have an

uncountably infinite

orthonormal set,
so this Hilbert space is not separable.

It is important not to confuse what we are doing here with the Fourier
integral. In that case, we are dealing with

L

2

, the functions

sin

⁡

a

⁢

x

have

infinite

L

2

norm (so they are not elements of that
Hilbert space) and the expansion of a function in terms of them is a

direct integral

. By contrast, in the case propounded here, the
expansion of a function of this space in terms of them would take the
form of a

direct sum

, just as with the Fourier series of a function on
a finite

interval

.

Title

example of non-separable Hilbert space

Canonical name

ExampleOfNonseparableHilbertSpace

Date of creation

2013-03-22 15:44:25

Last modified on

2013-03-22 15:44:25

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Example

Classification

msc 46C05

Related topic

AlmostPeriodicFunction

Generated on Fri Feb 9 21:38:12 2018 by

LaTeXML
generalization of the parallelogram law

generalization of the parallelogram law

Theorem

.

In an

inner product space

(

http://planetmath.org/InnerProductSpace

), let

x

,

y

,

z

be vectors. Then

∥

x

+

y

∥

2

+

∥

y

+

z

∥

2

+

∥

z

+

x

∥

2

=

∥

x

∥

2

+

∥

y

∥

2

+

∥

z

∥

2

+

∥

x

+

y

+

z

∥

2

.

Taking

x

+

z

=

0

we have the usual

parallelogram law

.

Title

generalization

of the parallelogram law

Canonical name

GeneralizationOfTheParallelogramLaw

Date of creation

2013-03-22 16:08:53

Last modified on

2013-03-22 16:08:53

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

10

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 46C05

Related topic

ParallelogramLaw2

Generated on Fri Feb 9 21:38:14 2018 by

LaTeXML
Hilbert module

Hilbert module

Definition 1

.

A

(right)

pre-Hilbert module

over a

C

*

-

algebra

A

is a right

A

-module

ℰ

equipped with an

A

-valued

inner product

⟨

-

,

-

⟩

:

ℰ

×

ℰ

→

A

,
i.e. a sesquilinear

pairing

satisfying

⟨

u

,

v

⁢

a

⟩

=

⟨

u

,

v

⟩

⁢

a

(1)

⟨

u

,

v

⟩

=

⟨

v

,

u

⟩

*

(2)

⟨

v

,

v

⟩

≥

0

,

with

⁢

⟨

v

,

v

⟩

=

0

⁢

iff

⁢

v

=

0

,

(3)

for all

u

,

v

∈

ℰ

and

a

∈

A

.
Note, positive definiteness is well-defined due to the notion of positivity for

C

*

-algebras.
The norm of an

element

v

∈

ℰ

is defined by

∥

v

∥

=

∥

⟨

v

,

v

⟩

∥

.

Definition 2

.

A

(right) Hilbert module

over a

C

*

-algebra

A

is a right pre-Hilbert module over

A

which is

complete

with respect to the norm.

Example 1

(Hilbert spaces)

A complex

Hilbert space

is a Hilbert

C

-module.

Example 2

(

C

*

-algebras)

A

C

*

-algebra

A

is a Hilbert

A

-module with inner product

⟨

a

,

b

⟩

=

a

*

⁢

b

.

Definition 3

.

A

Hilbert

A

-

B

-bimodule

is a (right) Hilbert module

ℰ

over a

C

*

-algebra

B

together with a *-

homomorphism

π

from a

C

*

-algebra

A

to

End

(

ℰ

)

.

Title

Hilbert module

Canonical name

HilbertModule

Date of creation

2013-03-22 13:01:01

Last modified on

2013-03-22 13:01:01

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

8

Author

mhale (572)

Entry type

Definition

Classification

msc 46C05

Synonym

C

*

-module

Related topic

HilbertSpace

Related topic

FinitelyGeneratedProjectiveModule

Defines

pre-Hilbert module

Generated on Fri Feb 9 21:38:16 2018 by

LaTeXML
Hilbert parallelotope

Hilbert parallelotope

The

Hilbert parallelotope

I

ω

is a

closed subset

of the

Hilbert space

ℝ

⁢

ℓ

2

(The symbol ’

ℝ

’ has been prefixed to indicate that the field of scalars is

ℝ

.) defined as

I

ω

=

{

(

a

0

,

a

1

,

a

2

,

…

)

∣

0

≤

a

i

≤

1

/

(

i

+

1

)

}

As a

topological space

,

I

ω

is

homeomorphic

to the

product

of a

countably infinite

number of copies of the

closed interval

[

0

,

1

]

. By

Tychonoff

’s

theorem

, this product is

compact

, so the Hilbert parallelotope is a compact subset of Hilbert space. This fact also explains the notation

I

ω

.

The Hilbert parallelotope enjoys a remarkable universality

property

— every

second countable

metric space

is homeomorphic to a subset of the Hilbert parallelotope. Since

second countability is hereditary

, the

converse

is also true — every subset of the Hilbert parallelotope is a second countable metric space.

Title

Hilbert parallelotope

Canonical

name

HilbertParallelotope

Date of creation

2013-03-22 14:38:32

Last modified on

2013-03-22 14:38:32

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 46C05

Synonym

Hilbert cube

Generated on Fri Feb 9 21:38:18 2018 by

LaTeXML