aarjaneiro/planetmath huggingface dataset:
Hilbert space

Hilbert space

A

Hilbert space

is an

inner product space

which is

http://planetmath.org/node/603

complete

under the metric.

In particular, a Hilbert space is a

Banach space

in the norm by the

inner product

, since the norm and the inner product both induce the same metric. Any

finite-dimensional

inner product space is a Hilbert space, but it is worth mentioning that some authors require the space to be infinite dimensional for it to be called a Hilbert space.

Title

Hilbert space

Canonical name

HilbertSpace

Date of creation

2013-03-22 12:19:06

Last modified on

2013-03-22 12:19:06

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Definition

Classification

msc 46C05

Related topic

InnerProductSpace

Related topic

HilbertModule

Related topic

QuadraticFunctionAssociatedWithALinearFunctional

Related topic

VectorNorm

Related topic

RieszSequence

Related topic

VonNeumannAlgebra

Related topic

HilbertSpacesAndQuantumGroupsVonNeumannAlgebras

Related topic

L2SpacesAreHilbertSpaces

Related topic

QuantumGroupsAndVonNeumannAlgebras

Related topic

HAlgebra

Related topic

Ries

Generated on Fri Feb 9 21:38:20 2018 by

LaTeXML
Hlawka’s inequality

Hlawka’s inequality

Theorem 1

In an

inner product space

(

http://planetmath.org/InnerProductSpace

), let

x

,

y

,

z

be vectors. Then

∥

x

+

y

∥

+

∥

y

+

z

∥

+

∥

z

+

x

∥

≤

∥

x

∥

+

∥

y

∥

+

∥

z

∥

+

∥

x

+

y

+

z

∥

.

Title

Hlawka’s

inequality

Canonical name

HlawkasInequality

Date of creation

2013-03-22 16:08:56

Last modified on

2013-03-22 16:08:56

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 46C05

Generated on Fri Feb 9 21:38:22 2018 by

LaTeXML
L^2-spaces are Hilbert spaces

L

2

-spaces are Hilbert spaces

Let

(

X

,

𝔅

,

μ

)

be a

measure space

. Let

L

2

⁢

(

X

)

denote the

L

2

-space (

http://planetmath.org/LpSpace

) associated with this measure space, i.e.

L

2

⁢

(

X

)

consists of

measurable functions

f

:

X

⟶

ℂ

such that

∥

f

∥

2

:=

(

∫

X

|

f

|

2

⁢

𝑑

μ

)

1

2

<

∞

identified up to equivalence almost everywhere.

It is known that all

L

p

-spaces (

http://planetmath.org/LpSpace

), with

1

≤

p

≤

∞

, are

Banach spaces

with respect to the

L

p

-norm (

http://planetmath.org/LpSpace

)

∥

⋅

∥

p

. For

L

2

⁢

(

X

)

we can say more:

Theorem

-

L

2

⁢

(

X

)

is an

Hilbert Space

with respect to the

inner product

⟨

⋅

,

⋅

⟩

defined by

⟨

f

,

g

⟩

=

∫

X

f

⁢

g

¯

⁢

𝑑

μ

Proof:

Sesquilinearity

follows from the linearity of the

Lebesgue integral

(

http://planetmath.org/PropertiesOfTheLebesgueIntegralOfLebesgueIntegrableFunctions

) (that is, the inner product defined above is linear in the first

argument

and

conjugate

linear in the second one). The conjugate

symmetry

is evident.

Positive definiteness holds by construction: If

∫

X

|

f

|

2

⁢

𝑑

μ

=

0

, then

|

f

|

2

(and therefore

f

) is zero almost everywhere, thus the

equivalence class

of

f

is the equivalence class of the zero function (which is the

additive

neutral element of the space).

Completeness is proved for the general case of

L

p

-spaces in this article (

http://planetmath.org/ProofThatLpSpacesAreComplete

).

□

0.0.1

Remarks

•

The spaces

ℂ

n

or

ℝ

n

with the usual inner product are particular examples of

L

2

⁢

(

X

)

, choosing

X

=

{

1

,

…

,

n

}

with the

counting measure

.

•

Choosing appropriate spaces

X

it can be shown that all Hilbert spaces are

isometrically isomorphic

to a

L

2

-space.

Title

L

2

-spaces are Hilbert spaces

Canonical name

L2spacesAreHilbertSpaces

Date of creation

2013-03-22 17:32:25

Last modified on

2013-03-22 17:32:25

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

23

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46C05

Synonym

square integrable

functions

form an Hilbert space

Related topic

LpSpace

Related topic

HilbertSpace

Related topic

MeasureSpace

Related topic

BanachSpace

Related topic

RieszFischerTheorem

Defines

linear space of square integrable functions

Defines

sequilinearity

Generated on Fri Feb 9 21:38:24 2018 by

LaTeXML
orthonormal basis

orthonormal basis

Definition

An

orthonormal basis

(or

Hilbert basis

)
of an

inner product space

V

is a subset

B

of

V

satisfying the following two

properties

:

•

B

is an

orthonormal set

.

•

The

linear span

of

B

is dense in

V

.

The first condition means that all elements of

B

have norm

1

and every element of

B

is

orthogonal

(

http://planetmath.org/OrthogonalVectors

) to every other element of

B

.
The second condition says that every element of

V

can be approximated arbitrarily closely by (finite)

linear combinations

of elements of

B

.

Orthonormal bases of Hilbert spaces

Every

Hilbert space

has an orthonormal basis.
The cardinality of this orthonormal basis
is called the

dimension

of the Hilbert space.
(This is

well-defined

,
as the cardinality does not depend on the choice of orthonormal basis.
This dimension is not in general the same as
the usual concept of dimension for

vector spaces

(

http://planetmath.org/Dimension2

).)

If

B

is an orthonormal basis of a Hilbert space

H

,
then for every

x

∈

H

we have

x

=

∑

b

∈

B

⟨

x

,

b

⟩

⁢

b

.

Thus

x

is expressed as a (possibly infinite)
“linear combination” of elements of

B

.
The

expression

is well-defined,
because only countably many of the terms

⟨

x

,

b

⟩

⁢

b

are non-zero
(even if

B

itself is uncountable),
and if there are infinitely many non-zero terms
the series is unconditionally convergent.
For any

x

,

y

∈

H

we also have

⟨

x

,

y

⟩

=

∑

b

∈

B

⟨

x

,

b

⟩

⁢

⟨

b

,

y

⟩

.

Title

orthonormal basis

Canonical

name

OrthonormalBasis

Date of creation

2013-03-22 14:02:29

Last modified on

2013-03-22 14:02:29

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

19

Author

yark (2760)

Entry type

Definition

Classification

msc 46C05

Synonym

Hilbert basis

Related topic

RieszSequence

Related topic

Orthonormal

Related topic

ClassificationOfHilbertSpaces

Defines

dimension of a Hilbert space

Defines

dimension

Generated on Fri Feb 9 21:38:26 2018 by

LaTeXML
parallelogram law

parallelogram law

Theorem 1

In an

inner product space

(

http://planetmath.org/InnerProductSpace

), let

x

and

y

be vectors. Then

∥

x

+

y

∥

2

+

∥

x

-

y

∥

2

=

2

⁢

∥

x

∥

2

+

2

⁢

∥

y

∥

2

.

Title

parallelogram law

Canonical name

ParallelogramLaw1

Date of creation

2013-03-22 16:08:02

Last modified on

2013-03-22 16:08:02

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

9

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 46C05

Related topic

ParallelogramLaw

Related topic

GeneralizationOfTheParallelogramLaw

Generated on Fri Feb 9 21:38:28 2018 by

LaTeXML
polarization identity

polarization identity

Theorem

[

polarization identity

]

-

Let

X

be an

inner product space

over

ℝ

. The following identity holds for every

x

,

y

∈

X

:

⟨

x

,

y

⟩

=

1

4

⁢

(

∥

x

+

y

∥

2

-

∥

x

-

y

∥

2

)

If

X

is an inner product space over

ℂ

instead, the identity becomes

⟨

x

,

y

⟩

=

1

4

⁢

(

∥

x

+

y

∥

2

-

∥

x

-

y

∥

2

)

+

1

4

⁢

i

⁢

(

∥

x

+

i

⁢

y

∥

2

-

∥

x

-

i

⁢

y

∥

2

)

Remark -

This result shows that the

inner product

of

X

is determined by the norm. Moreover, it can be shown that if a

normed space

V

the

parallelogram law

, the above formulas define an inner product

compatible

with the norm of

V

.

Title

polarization identity

Canonical name

PolarizationIdentity

Date of creation

2013-03-22 17:37:20

Last modified on

2013-03-22 17:37:20

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 21:38:30 2018 by

LaTeXML
proof of Bessel inequality

proof of Bessel inequality

Let

r

n

=

x

-

∑

k

=

1

n

⟨

x

,

e

k

⟩

⋅

e

k

.

Then for

j

=

1

,

…

,

n

,

⟨

r

n

,

e

j

⟩

=

⟨

x

,

e

j

⟩

-

∑

k

=

1

n

⟨

⟨

x

,

e

k

⟩

⋅

e

k

,

e

j

⟩

(1)

=

⟨

x

,

e

j

⟩

-

⟨

x

,

e

j

⟩

⁢

⟨

e

j

,

e

j

⟩

=

0

(2)

so

e

1

,

…

,

e

n

,

r

n

is an orthogonal series.

Computing norms, we see that

∥

x

∥

2

=

∥

r

n

+

∑

k

=

1

n

⟨

x

,

e

k

⟩

⋅

e

k

∥

2

=

∥

r

n

∥

2

+

∑

k

=

1

n

|

⟨

x

,

e

k

⟩

|

2

≥

∑

k

=

1

n

|

⟨

x

,

e

k

⟩

|

2

.

So the series

∑

k

=

1

∞

|

⟨

x

,

e

k

⟩

|

2

converges and is

bounded

by

∥

x

∥

2

, as required.

Title

proof of Bessel inequality

Canonical name

ProofOfBesselInequality

Date of creation

2013-03-22 12:46:41

Last modified on

2013-03-22 12:46:41

Owner

ariels (338)

Last modified by

ariels (338)

Numerical id

4

Author

ariels (338)

Entry type

Proof

Classification

msc 46C05

Generated on Fri Feb 9 21:38:32 2018 by

LaTeXML
proof of generalization of the parallelogram law

proof of generalization of the parallelogram law

Let

g

⁢

(

x

,

y

)

=

∥

x

+

y

∥

2

-

∥

x

∥

2

and

m

⁢

(

x

,

y

)

=

⟨

x

,

y

⟩

+

⟨

y

,

x

⟩

.
Then

g

⁢

(

x

,

y

)

=

∥

y

∥

2

+

m

⁢

(

x

,

y

)

.

Hence, taking

x

1

=

x

4

=

x

,

x

2

=

y

,

x

3

=

z

we have:

∑

i

=

1

3

∥

x

i

+

x

i

+

1

∥

2

-

∑

i

=

1

3

∥

x

i

∥

2

=

∑

i

=

1

3

g

⁢

(

x

i

,

x

i

+

1

)

=

∑

i

=

1

3

∥

x

i

∥

2

+

∑

i

=

1

3

m

⁢

(

x

i

,

x

i

+

1

)

=

∥

∑

i

=

1

3

x

i

∥

2

.

Title

proof of

generalization

of the

parallelogram law

Canonical name

ProofOfGeneralizationOfTheParallelogramLaw

Date of creation

2013-03-22 16:08:58

Last modified on

2013-03-22 16:08:58

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Proof

Classification

msc 46C05

Generated on Fri Feb 9 21:38:34 2018 by

LaTeXML
proof of parallelogram law

proof of parallelogram law

The proof supplied here for the

parallelogram law

uses the properties of norms and

inner products

. See the entries about these for more details regarding the following calculations.

Proof.

∥

x

+

y

∥

2

+

∥

x

-

y

∥

2

=

⟨

x

+

y

,

x

+

y

⟩

+

⟨

x

-

y

,

x

-

y

⟩

=

⟨

x

,

x

+

y

⟩

+

⟨

y

,

x

+

y

⟩

+

⟨

x

,

x

-

y

⟩

-

⟨

y

,

x

-

y

⟩

=

⟨

x

+

y

,

x

⟩

¯

+

⟨

x

+

y

,

y

⟩

¯

+

⟨

x

-

y

,

x

⟩

¯

-

⟨

x

-

y

,

y

⟩

¯

=

⟨

x

,

x

⟩

+

⟨

y

,

x

⟩

¯

+

⟨

x

,

y

⟩

+

⟨

y

,

y

⟩

¯

+

⟨

x

,

x

⟩

-

⟨

y

,

x

⟩

¯

-

(

⟨

x

,

y

⟩

-

⟨

y

,

y

⟩

¯

)

=

⟨

x

,

x

⟩

¯

+

⟨

y

,

x

⟩

¯

+

⟨

x

,

y

⟩

¯

+

⟨

y

,

y

⟩

¯

+

⟨

x

,

x

⟩

¯

-

⟨

y

,

x

⟩

¯

-

⟨

x

,

y

⟩

¯

+

⟨

y

,

y

⟩

¯

=

⟨

x

,

x

⟩

+

⟨

y

,

y

⟩

+

⟨

x

,

x

⟩

+

⟨

y

,

y

⟩

=

2

⁢

⟨

x

,

x

⟩

+

2

⁢

⟨

y

,

y

⟩

=

2

⁢

∥

x

∥

2

+

2

⁢

∥

y

∥

2

.

∎

Title

proof of parallelogram law

Canonical name

ProofOfParallelogramLaw1

Date of creation

2013-03-22 16:08:15

Last modified on

2013-03-22 16:08:15

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 46C05

Related topic

ProofOfParallelogramLaw

Related topic

AlternateProofOfParallelogramLaw

Generated on Fri Feb 9 21:38:37 2018 by

LaTeXML
Pythagorean theorem in inner product spaces

Pythagorean theorem in inner product spaces

-

Let

X

be an

inner product space

(over

ℝ

or

ℂ

) and

x

,

y

∈

X

two

orthogonal vectors

. Then

∥

x

+

y

∥

2

=

∥

x

∥

2

+

∥

y

∥

2

.

Proof :

As

x

⟂

y

one has

⟨

x

,

y

⟩

=

0

. Then

∥

x

+

y

∥

2

=

⟨

x

+

y

,

x

+

y

⟩

=

⟨

x

,

x

⟩

+

⟨

x

,

y

⟩

+

⟨

y

,

x

⟩

+

⟨

y

,

y

⟩

=

∥

x

∥

2

+

⟨

x

,

y

⟩

+

⟨

x

,

y

⟩

¯

+

∥

y

∥

2

=

∥

x

∥

2

+

∥

y

∥

2

□

R

⁢

e

⁢

m

⁢

a

⁢

r

⁢

k

-

This theorem is valid (with the same proof) for spaces with a semi-definite

inner product

.

Title

Pythagorean theorem in inner product spaces

Canonical name

PythagoreanTheoremInInnerProductSpaces

Date of creation

2013-03-22 17:32:13

Last modified on

2013-03-22 17:32:13

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46C05

Synonym

Pythagoras theorem in inner product spaces

Related topic

PythagorasTheorem

Generated on Fri Feb 9 21:38:39 2018 by

LaTeXML
Rellich selection theorem

Rellich selection theorem

Let

D

be an open subset of

ℝ

n

. If, for a sequence of functions

f

i

:

D

→

ℝ

,

i

=

1

,

2

,

…

there exists a constant

B

>

0

such that

(

∀

i

)

∥

f

i

∥

L

2

⁢

(

D

)

=

∫

D

f

i

2

⁢

d

n

⁢

x

<

B

and

(

∀

i

)

(

∀

j

∈

{

1

,

…

n

}

)

∫

D

(

∂

⁡

f

i

∂

⁡

x

j

)

2

d

n

x

<

B

then there exists a

subsequence

which is convergent in the

L

2

⁢

(

D

)

norm.

Title

Rellich selection theorem

Canonical

name

RellichSelectionTheorem

Date of creation

2013-03-22 14:38:55

Last modified on

2013-03-22 14:38:55

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 46C05

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

LaTeXML
Riesz sequence

Riesz sequence

Definition

A sequence of vectors

(

x

n

)

in a

Hilbert space

H

is called a

Riesz sequence

if there exist constants

0

<

c

≤

C

such that

c

⁢

(

∑

n

|

a

n

|

2

)

≤

∥

∑

n

a

n

⁢

x

n

∥

2

≤

C

⁢

(

∑

n

|

a

n

|

2

)

for all sequences of scalars

(

a

n

)

∈

l

2

. A Riesz sequence is called a Riesz basis if

span

(

x

n

)

¯

=

H

.

If

H

is a finite-dimensional space, then every basis of

H

is a Riesz basis.

Title

Riesz sequence

Canonical name

RieszSequence

Date of creation

2013-03-22 14:26:53

Last modified on

2013-03-22 14:26:53

Owner

swiftset (1337)

Last modified by

swiftset (1337)

Numerical id

7

Author

swiftset (1337)

Entry type

Definition

Classification

msc 46C05

Related topic

OrthonormalBasis

Related topic

HilbertSpace

Related topic

Frame2

Related topic

AnEquivalentConditionForTheTranslatesOfAnL_2FunctionToFormARieszBasis

Defines

Riesz basis

Generated on Fri Feb 9 21:38:43 2018 by

LaTeXML
Schur’s condition for a matrix to be a bounded operator on l^2

Schur’s condition for a matrix to be a bounded operator on

l

2

Theorem 0.1

Let

B

be a matrix defined on

T

×

T

for some

countable set

T

. If there exists a positive number

C

such that

∑

t

∈

T

|

b

⁢

(

s

,

t

)

|

<

C

⁢

for all

⁢

s

𝑎𝑛𝑑

∑

s

∈

T

|

b

⁢

(

s

,

t

)

|

<

C

⁢

for all

⁢

t

,

then

B

is a

bounded operator

on

l

2

⁢

(

T

)

with its

operator norm

∥

B

∥

less than or equal to

C

.

Proof.

Let

x

be a

sequence

in

l

2

⁢

(

T

)

. We have

∥

B

⁢

x

∥

2

2

=

∑

s

∈

T

|

∑

t

∈

T

b

⁢

(

s

,

t

)

⁢

x

⁢

(

t

)

|

2

≤

∑

s

∈

T

[

∑

t

∈

T

|

b

⁢

(

s

,

t

)

|

⁢

(

|

b

⁢

(

s

,

t

)

|

⁢

|

x

⁢

(

t

)

|

)

]

2

≤

∑

s

∈

T

[

(

∑

t

∈

T

|

b

⁢

(

s

,

t

)

|

)

⁢

(

∑

t

∈

T

|

b

⁢

(

s

,

t

)

|

⁢

|

x

⁢

(

t

)

|

2

)

]

≤

C

⁢

∑

s

∈

T

∑

t

∈

T

|

b

⁢

(

s

,

t

)

|

⁢

|

x

⁢

(

t

)

|

2

≤

C

⁢

∑

t

∈

T

|

x

⁢

(

t

)

|

2

⁢

∑

s

∈

T

|

b

⁢

(

s

,

t

)

|

≤

C

2

⁢

∑

t

∈

T

|

x

⁢

(

t

)

|

2

.

Therefore we have

∥

B

⁢

x

∥

2

≤

C

⁢

∥

x

∥

2

for all

x

∈

l

2

⁢

(

T

)

, hence

∥

B

∥

≤

C

.

□

Title

Schur’s condition for a matrix to be a bounded operator on

l

2

Canonical name

SchursConditionForAMatrixToBeABoundedOperatorOnL2

Date of creation

2013-03-22 15:57:18

Last modified on

2013-03-22 15:57:18

Owner

Gorkem (3644)

Last modified by

Gorkem (3644)

Numerical id

5

Author

Gorkem (3644)

Entry type

Theorem

Classification

msc 46C05

Synonym

Schur’s Lemma

Synonym

Schur’s Lemma for infinite matrices

Generated on Fri Feb 9 21:38:46 2018 by

LaTeXML
yet another proof of parallelogram law

yet another proof of parallelogram law

Define

g

⁢

(

ϵ

)

=

⟨

x

+

ϵ

⁢

y

,

x

+

ϵ

⁢

y

⟩

, where

ϵ

is real.
Then

g

⁢

(

ϵ

)

=

⟨

x

,

x

⟩

+

ϵ

⁢

(

⟨

y

,

x

⟩

+

⟨

x

,

y

⟩

)

+

ϵ

2

⁢

⟨

y

,

y

⟩

.

Hence,

∥

x

+

y

∥

2

+

∥

x

-

y

∥

2

=

g

⁢

(

1

)

+

g

⁢

(

-

1

)

=

2

⁢

⟨

x

,

x

⟩

+

2

⁢

⟨

y

,

y

⟩

=

2

⁢

∥

x

∥

2

+

2

⁢

∥

y

∥

2

.

Title

yet another

proof of parallelogram law

Canonical name

YetAnotherProofOfParallelogramLaw

Date of creation

2013-03-22 16:08:23

Last modified on

2013-03-22 16:08:23

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

5

Author

Mathprof (13753)

Entry type

Proof

Classification

msc 46C05

Generated on Fri Feb 9 21:38:48 2018 by

LaTeXML
ERROR(\delimiter)⁢"⁢026⁢A⁢30⁢C⁢ERROR(\delimiter)⁢"⁢426830⁢A⁢T⁢v,v⁢ERROR(\delimiter)⁢"⁢526930⁢B⁢ERROR(\delimiter)⁢"⁢026⁢A⁢30⁢C≤μ⁢ERROR(\delimiter)⁢"⁢026⁢B⁢30⁢D⁢v⁢ERROR(\delimiter)⁢"⁢026⁢B⁢30⁢D^2 for all v implies ERROR(\delimiter)⁢"⁢026⁢B⁢30⁢D⁢T⁢ERROR(\delimiter)⁢"⁢026⁢B⁢30⁢D≤μ

\delimiter

⁢

"

⁢

026

⁢

A

⁢

30

⁢

C

⁢

\delimiter

⁢

"

⁢

426830

⁢

A

⁢

T

⁢

v

,

v

⁢

\delimiter

⁢

"

⁢

526930

⁢

B

⁢

\delimiter

⁢

"

⁢

026

⁢

A

⁢

30

⁢

C

≤

μ

⁢

\delimiter

⁢

"

⁢

026

⁢

B

⁢

30

⁢

D

⁢

v

⁢

\delimiter

⁢

"

⁢

026

⁢

B

⁢

30

⁢

D

2

for all

v

implies

\delimiter

⁢

"

⁢

026

⁢

B

⁢

30

⁢

D

⁢

T

⁢

\delimiter

⁢

"

⁢

026

⁢

B

⁢

30

⁢

D

≤

μ

Theorem

.

Let

H

be a

unitary space

,

T

be a

self-adjoint

linear operator

and

μ

≥

0

. If

|

⟨

T

⁢

v

,

v

⟩

|

≤

μ

⁢

∥

v

∥

2

for all

v

∈

H

then

T

is a

bounded operator

and

∥

T

∥

≤

μ

.

Proof.

We will show that

∥

T

⁢

v

∥

≤

μ

⁢

∥

v

∥

for all

v

∈

H

.
This is trivial if

∥

T

⁢

v

∥

or

∥

v

∥

is zero, so assume
they are not. Let

λ

be any positive number.

∥

T

⁢

v

∥

2

=

⟨

T

⁢

v

,

T

⁢

v

⟩

=

1

4

⁢

[

⟨

T

⁢

(

λ

⁢

v

+

1

λ

⁢

T

⁢

v

)

,

(

λ

⁢

v

+

1

λ

⁢

T

⁢

v

)

⟩

-

⟨

T

⁢

(

λ

⁢

v

-

1

λ

⁢

T

⁢

v

)

,

(

λ

⁢

v

-

1

λ

⁢

T

⁢

v

)

⟩

]

≤

μ

4

⁢

[

∥

λ

⁢

v

+

1

λ

⁢

T

⁢

v

∥

2

+

∥

λ

⁢

v

-

1

λ

⁢

T

⁢

v

∥

2

]

≤

μ

2

⁢

[

λ

2

⁢

∥

v

∥

2

+

1

λ

2

⁢

∥

T

⁢

v

∥

2

]

Now if we put

λ

2

=

∥

T

⁢

v

∥

∥

v

∥

we get

∥

T

⁢

v

∥

2

≤

μ

⁢

∥

T

⁢

v

∥

⁢

∥

v

∥

hence

∥

T

⁢

v

∥

≤

μ

⁢

∥

v

∥

.
∎

Reference:

F. Riesz and B. Sz-Nagy,

Functional Analysis

, F. Ungar
Publishing, 1955, chap VI.

Title

\delimiter

⁢

"

⁢

026

⁢

A

⁢

30

⁢

C

⁢

\delimiter

⁢

"

⁢

426830

⁢

A

⁢

T

⁢

v

,

v

⁢

\delimiter

⁢

"

⁢

526930

⁢

B

⁢

\delimiter

⁢

"

⁢

026

⁢

A

⁢

30

⁢

C

≤

μ

⁢

\delimiter

⁢

"

⁢

026

⁢

B

⁢

30

⁢

D

⁢

v

⁢

\delimiter

⁢

"

⁢

026

⁢

B

⁢

30

⁢

D

2

for all

v

implies

\delimiter

⁢

"

⁢

026

⁢

B

⁢

30

⁢

D

⁢

T

⁢

\delimiter

⁢

"

⁢

026

⁢

B

⁢

30

⁢

D

≤

μ

Canonical

name

vertlangleTvvranglevertleqmuVertVVert2ForAllVImpliesVertTVertleqmu

Date of creation

2013-03-22 15:25:33

Last modified on

2013-03-22 15:25:33

Owner

Gorkem (3644)

Last modified by

Gorkem (3644)

Numerical id

16

Author

Gorkem (3644)

Entry type

Theorem

Classification

msc 46C05

Generated on Fri Feb 9 21:38:51 2018 by

LaTeXML
projections and closed subspaces

projections and closed subspaces

Theorem

1 -

Let

X

be a

Banach space

and

M

a closed subspace. Then,

•

M

is

topologically complemented

in

X

if and only if there exists a

continuous

projection

onto

M

.

•

Given a topological complement

N

of

M

, there exists a unique continuous projection

P

onto

M

such that

P

⁢

(

x

+

y

)

=

x

for all

x

∈

M

and

y

∈

N

.

The projection

P

in the second part of the above theorem is sometimes called the

projection onto

M

along

N

.

The above result can be further improved for

Hilbert spaces

.

Theorem 2 -

Let

X

be a Hilbert space and

M

a closed subspace. Then,

M

is topologically complemented in

X

if and only if there exists an orthogonal projection onto

M

(which is unique).

Since, by the

orthogonal decomposition theorem

, a closed subspace of a Hilbert space is always topologically complemented by its

orthogonal complement

(

X

=

M

⊕

M

⟂

), it follows that

Corollary -

Let

X

be a Hilbert space and

M

a closed subspace. Then, there exists a unique orthogonal projection onto

M

. This establishes a

bijective

correspondence between orthogonal projections and closed subspaces.

Title

projections and closed subspaces

Canonical name

ProjectionsAndClosedSubspaces

Date of creation

2013-03-22 17:52:57

Last modified on

2013-03-22 17:52:57

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46C07

Classification

msc 46B20

Synonym

projection along a closed subspace

Synonym

orthogonal projections onto Hilbert subspaces

Generated on Fri Feb 9 21:38:57 2018 by

LaTeXML
Banach spaces with complemented subspaces

Banach spaces with complemented subspaces

Theorem. [Lindenstrauss-Tzafriri]

Let

V

be a

Banach space

, such that for each closed subspace

M

there exists a closed subspace

N

such that

M

∩

N

=

0

and

M

+

N

=

V

(i.e. every closed subspace is

complemented

). Then

V

is

isomorphic

to a

Hilbert space

(i.e. there exists a Hilbert space structure on

V

that induces the original

topology

on

V

as a Banach space).

Title

Banach spaces with complemented subspaces

Canonical name

BanachSpacesWithComplementedSubspaces

Date of creation

2013-03-22 16:02:59

Last modified on

2013-03-22 16:02:59

Owner

aube (13953)

Last modified by

aube (13953)

Numerical id

13

Author

aube (13953)

Entry type

Theorem

Classification

msc 46C15

Synonym

Lindenstrauss-Tzafriri theorem

Synonym

Lindenstrauss-Tzafriri complemented subspace theorem

Generated on Fri Feb 9 21:39:01 2018 by

LaTeXML
classification of separable Hilbert spaces

classification of separable Hilbert spaces

Let

H

1

and

H

2

be

infinite dimensional

, separable Hilbert spaces. Then there is an

isomorphism

f

:

H

1

→

H

2

which is also an isometry.

In other words,

H

1

and

H

2

are identical as

Hilbert spaces

.

Title

classification of separable Hilbert spaces

Canonical name

ClassificationOfSeparableHilbertSpaces

Date of creation

2013-03-22 12:19:12

Last modified on

2013-03-22 12:19:12

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

5

Author

Evandar (27)

Entry type

Theorem

Classification

msc 46C15

Related topic

ClassificationOfHilbertSpaces

Generated on Fri Feb 9 21:39:05 2018 by

LaTeXML