aarjaneiro/planetmath huggingface dataset:
Hahn-Banach theorem

Hahn-Banach theorem

The

Hahn-Banach theorem

is a foundational result in

functional
analysis

. Roughly speaking, it asserts the

existence

of a great

variety

of bounded (and hence

continuous

)

linear functionals

on an

normed vector space

, even if that space happens to be

infinite-dimensional

. We first consider an
abstract version of this

theorem

, and then give the more classical
result as a corollary.

Let

V

be a real, or a

complex vector space

, with

K

denoting the corresponding field of scalars, and let

p

:

V

→

ℝ

+

be a

seminorm

on

V

.

Theorem 1

Let

f

:

U

→

K

be a linear functional defined on a

subspace

U

⊂

V

. If the restricted

functional

satisfies

|

f

⁢

(

𝐮

)

|

≤

p

⁡

(

𝐮

)

,

𝐮

∈

U

,

then it can be extended to all of

V

without violating the above

property

. To be more precise, there exists a linear functional

F

:

V

→

K

such that

F

⁢

(

𝐮

)

=

f

⁢

(

𝐮

)

,

𝐮

∈

U

|

F

⁢

(

𝐮

)

|

≤

p

⁡

(

𝐮

)

,

𝐮

∈

V

.

Definition 2

We say that a linear functional

f

:

V

→

K

is

bounded

if
there exists a bound

B

∈

R

+

such that

|

f

⁢

(

𝐮

)

|

≤

B

⁢

p

⁡

(

𝐮

)

,

𝐮

∈

V

.

(1)

If

f

is a bounded linear functional, we define

∥

f

∥

, the
norm of

f

, according to

∥

f

∥

=

sup

⁡

{

|

f

⁢

(

𝐮

)

|

:

p

⁡

(

𝐮

)

=

1

}

.

One can show that

∥

f

∥

is the

infimum

of all the possible

B

that satisfy (

1

)

Theorem 3

(Hahn-Banach)

Let

f

:

U

→

K

be a bounded linear functional defined on a subspace

U

⊂

V

. Let

∥

f

∥

U

denote the norm of

f

relative
to the restricted seminorm on

U

. Then there exists a bounded

extension

F

:

V

→

K

with the same norm, i.e.

∥

F

∥

V

=

∥

f

∥

U

.

Title

Hahn-Banach theorem

Canonical

name

HahnBanachTheorem

Date of creation

2013-03-22 12:54:09

Last modified on

2013-03-22 12:54:09

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

10

Author

rmilson (146)

Entry type

Theorem

Classification

msc 46B20

Defines

bound

Defines

bounded

Generated on Fri Feb 9 21:36:25 2018 by

LaTeXML
Minkowski functional

Minkowski functional

Let

X

be a

normed space

and let

K

an

absorbing

convex subset

of

X

such that

0

is in the interior of

K

.
Then the

Minkowski functional

ρ

:

X

→

ℝ

is defined as

ρ

⁢

(

x

)

=

inf

⁡

{

λ

>

0

:

x

∈

λ

⁢

K

}

.

We put

ρ

⁢

(

x

)

=

0

whenever

x

=

0

. Clearly

ρ

⁢

(

x

)

≥

0

for all

x

.

It is important to note that in general

ρ

⁢

(

x

)

≠

ρ

⁢

(

-

x

)

.

Properties

ρ

is positively

1

-

homogeneous

. This means that

ρ

⁢

(

s

⋅

x

)

=

s

⋅

ρ

⁢

(

x

)

for

s

>

0

.

Title

Minkowski functional

Canonical name

MinkowskiFunctional

Date of creation

2013-03-22 14:50:44

Last modified on

2013-03-22 14:50:44

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

18

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 46B20

Generated on Fri Feb 9 21:36:27 2018 by

LaTeXML
normed plane

normed plane

A

normed plane

is a pair

(

ℝ

2

,

|

|

⋅

|

|

)

, where the function

x

→

||

x

||

is a norm.

If we define a distance function

d

⁢

(

x

,

y

)

=

||

x

-

y

||

then
the

metric space

(

ℝ

2

,

d

)

is called a

Minkowski plane

or
a

Minkowski geometry

.

The classical examples of Minkowski and normed planes are
the

p

-norm

||

x

||

p

=

(

|

x

1

|

p

+

|

x

2

|

p

)

1

/

p

where

1

≤

p

<

∞

and the maximum or

supremum norm

||

x

||

∞

=

max

⁡

{

|

x

1

|

,

|

x

2

|

}

.

Title

normed plane

Canonical name

NormedPlane

Date of creation

2013-03-22 16:50:30

Last modified on

2013-03-22 16:50:30

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 46B20

Defines

Minkowski plane

Defines

Minkowski geometry

Generated on Fri Feb 9 21:36:29 2018 by

LaTeXML
nuclear space

nuclear space

If

E

is a FrÃÂ©chet space and

(

p

j

)

an

increasing sequence

of

semi-norms

on

E

defining the

topology

of

E

, we have

E

=

lim

⟵

⁢

E

^

p

j

,

where

E

^

p

j

is the Hausdorff completion of

(

E

,

p

j

)

and

E

^

p

j

+

1

→

E

^

p

j

the

canonical

morphism. Here

E

^

p

j

is a

Banach space

for the

induced norm

p

^

j

.

A FrÃÂ©chet space

E

is said to be

nuclear

if the topology of

E

can be defined by an increasing sequence of semi-norms

p

j

such that each canonical morphism

E

^

p

j

+

1

→

E

^

p

j

of Banach spaces is nuclear.

Recall that a morphism

f

:

E

→

F

of

complete

locally convex spaces is said to be nuclear if

f

can be written as

f

⁢

(

x

)

=

∑

λ

j

⁢

ξ

j

⁢

(

x

)

⁢

y

j

where

(

λ

j

)

is a

sequence

of scalars with

∑

|

λ

j

|

<

+

∞

,

ξ

j

∈

E

′

an equicontinuous sequence of

linear forms

and

y

j

∈

F

a

bounded

sequence.

Title

nuclear space

Canonical name

NuclearSpace

Date of creation

2013-03-22 16:37:28

Last modified on

2013-03-22 16:37:28

Owner

Simone (5904)

Last modified by

Simone (5904)

Numerical id

6

Author

Simone (5904)

Entry type

Definition

Classification

msc 46B20

Generated on Fri Feb 9 21:36:31 2018 by

LaTeXML
properties of Minkowski’s functional

properties of Minkowski’s functional

Let

X

be a

normed space

,

K

convex subset of

X

and

0

belongs to the

interior

of

K

.Then

1.

ρ

K

⁢

(

x

)

≥

0

for all

x

∈

X

2.

ρ

K

⁢

(

0

)

=

0

3.

ρ

K

⁢

(

λ

⁢

x

)

=

λ

⁢

ρ

K

⁢

(

x

)

, for all

λ

≥

0

and

x

∈

X

4.

ρ

K

⁢

(

x

+

y

)

≤

ρ

K

⁢

(

x

)

+

ρ

K

⁢

(

y

)

for all

x

,

y

∈

K

5.

{

x

∈

X

:

ρ

K

⁢

(

x

)

<

1

}

⊂

K

⊂

{

x

∈

X

:

ρ

K

⁢

(

x

)

≤

1

}

6.

K

0

=

{

x

∈

X

:

ρ

K

⁢

(

x

)

<

1

}

where

K

0

denotes the interior of

K

7.

K

¯

=

{

x

∈

X

:

ρ

K

⁢

(

x

)

≤

1

}

where

K

¯

denotes the

closure

of

K

8.

B

⁢

d

⁢

(

K

)

=

{

x

∈

X

:

ρ

K

⁢

(

x

)

=

1

}

where the

B

⁢

d

⁢

(

K

)

denotes the

boundary

of

K

.

Minkowski’s

functional

is a useful tool to prove

propositions

and solve exercises. Let us see an example

Example

Let

K

be a convex subset of

X

. Show that

E

⁢

x

⁢

(

K

)

⊂

B

⁢

d

⁢

(

K

)

, where

E

⁢

x

⁢

(

K

)

denotes the
set of extreme points of

K

.

If

x

∈

E

⁢

x

⁢

(

K

)

then from this follows that

x

∈

1

⁢

K

and

ρ

K

⁢

(

x

)

=

1

.
Now we hypothesize that

ρ

K

⁢

(

x

)

<

1

then there is a

real number

s

such that

ρ

K

⁢

(

x

)

<

s

<

1

and
so

ρ

K

⁢

(

x

s

)

<

1

. Therefore we have that

x

=

s

⁢

x

s

+

(

1

-

s

)

⁢

0

∈

K

, that contradicts to the
fact that

x

∈

E

⁢

x

⁢

(

K

)

.

Title

properties

of Minkowski’s functional

Canonical

name

PropertiesOfMinkowskisFunctional

Date of creation

2013-03-22 15:45:04

Last modified on

2013-03-22 15:45:04

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

10

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 46B20

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

LaTeXML
subadditive

subadditive

Suppose

V

is a

vector space

(over a field), and

f

is a

function

f

:

V

→

ℝ

. Then

f

is

subadditive

if

f

⁢

(

x

+

y

)

≤

f

⁢

(

x

)

+

f

⁢

(

y

)

,

x

,

y

∈

V

.

Examples

1.

Any

linear function

V

→

ℝ

is subadditive.

2.

If

∥

⋅

∥

is a norm on

V

,

a

≥

0

, then

f

⁢

(

x

)

=

a

+

∥

x

∥

is subadditive.

Properties

Suppose

f

is subadditive.

1.

If

f

is positively

1

-

homogeneous

, then

f

is convex.

2.

The sum of two subadditive functions is subadditive.

Title

subadditive

Canonical name

Subadditive

Date of creation

2013-03-22 15:12:26

Last modified on

2013-03-22 15:12:26

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Definition

Classification

msc 46B20

Related topic

LinearTransformation

Related topic

Seminorm

Related topic

HomogeneousFunction

Generated on Fri Feb 9 21:36:42 2018 by

LaTeXML
vector norm

vector norm

A

vector norm

on the

real vector space

V

is a function

f

:

V

→

ℝ

that satisfies the following properties:

f

⁢

(

x

)

=

0

⇔

x

=

0

f

⁢

(

x

)

≥

0

x

∈

V

f

⁢

(

x

+

y

)

≤

f

⁢

(

x

)

+

f

⁢

(

y

)

x

,

y

∈

V

f

⁢

(

α

⁢

x

)

=

|

α

|

⁢

f

⁢

(

x

)

α

∈

ℝ

,

x

∈

V

Such a function is denoted as

||

x

||

. Particular norms are distinguished by subscripts, such
as

||

x

||

V

, when referring to a norm in the space

V

. A

unit vector

with respect to the norm

|

|

⋅

|

|

is a vector

x

satisfying

||

x

||

=

1

.

A vector norm on a

complex vector space

is defined similarly.

A common (and useful) example of a real norm is the

Euclidean norm

given by

||

x

||

=

(

x

1

2

+

x

2

2

+

⋯

+

x

n

2

)

1

/

2

defined on

V

=

ℝ

n

.
Note, however, that there exists

vector spaces

which are metric, but upon which it is not possible to define a norm. If it possible, the space is called a

normed vector space

. Given a metric on the vector space, a necessary and sufficient condition for this space to be a normed space, is

d

⁢

(

x

+

a

,

y

+

a

)

=

d

⁢

(

x

,

y

)

∀

x

,

y

,

a

∈

V

d

⁢

(

α

⁢

x

,

α

⁢

y

)

=

|

α

|

⁢

d

⁢

(

x

,

y

)

∀

x

,

y

∈

V

,

α

∈

ℝ

But given a norm, a metric can always be defined by the

equation

d

⁢

(

x

,

y

)

=

||

x

-

y

||

. Hence every normed space is a metric space.

Title

vector norm

Canonical name

VectorNorm

Date of creation

2013-03-22 11:43:00

Last modified on

2013-03-22 11:43:00

Owner

mike (2826)

Last modified by

mike (2826)

Numerical id

30

Author

mike (2826)

Entry type

Definition

Classification

msc 46B20

Classification

msc 18-01

Classification

msc 20H15

Classification

msc 20B30

Related topic

Vector

Related topic

Metric

Related topic

Norm

Related topic

VectorPnorm

Related topic

NormedVectorSpace

Related topic

MatrixNorm

Related topic

MatrixPnorm

Related topic

FrobeniusMatrixNorm

Related topic

CauchySchwarzInequality

Related topic

MetricSpace

Related topic

VectorSpace

Related topic

LpSpace

Related topic

OperatorNorm

Related topic

BoundedOperator

Related topic

SemiNorm

Related topic

BanachSpace

Related topic

HilbertSpace

Related topic

UnitVector

Defines

normed vector space

Defines

Euclidean norm

Generated on Fri Feb 9 21:36:44 2018 by

LaTeXML
vector p-norm

vector p-norm

A class of

vector norms

, called a

p

-norm and denoted

|

|

⋅

|

|

p

, is defined as

||

x

||

p

=

(

|

x

1

|

p

+

⋯

+

|

x

n

|

p

)

1

p

p

≥

1

,

x

∈

ℝ

n

The most widely used are the 1-norm, 2-norm, and

∞

-norm:

||

x

||

1

=

|

x

1

|

+

⋯

+

|

x

n

|

||

x

||

2

=

|

x

1

|

2

+

⋯

+

|

x

n

|

2

=

x

T

⁢

x

||

x

||

∞

=

max

1

≤

i

≤

n

⁡

|

x

i

|

The 2-norm is sometimes called the

Euclidean vector norm

, because

||

x

-

y

||

2

yields the

Euclidean distance

between any two vectors

x

,

y

∈

ℝ

n

. The 1-norm is also called the

taxicab metric

(sometimes Manhattan metric) since the

distance

of two points can be viewed as the distance a taxi would travel on a city (horizontal and vertical movements).

A useful fact is that for finite dimensional spaces (like

ℝ

n

) the three mentioned norms are

http://planetmath.org/node/4312

equivalent

. Moreover, all

p

-norms are equivalent. This can be proved using that any norm has to be

continuous

in the

2

-norm and working in the unit circle.

The

L

p

-norm (

http://planetmath.org/LpSpace

) in

function spaces

is a

generalization

of these norms by using

counting measure

.

Title

vector p-norm

Canonical

name

VectorPnorm

Date of creation

2013-03-22 11:43:03

Last modified on

2013-03-22 11:43:03

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

21

Author

Andrea Ambrosio (7332)

Entry type

Definition

Classification

msc 46B20

Classification

msc 05Cxx

Classification

msc 05-01

Classification

msc 20H15

Classification

msc 20B30

Synonym

Minkowski norm

Synonym

Euclidean vector norm

Synonym

vector Euclidean norm

Synonym

vector 1-norm

Synonym

vector 2-norm

Synonym

vector infinity-norm

Synonym

L^p metric

Synonym

L^p

Related topic

VectorNorm

Related topic

CauchySchwartzInequality

Related topic

HolderInequality

Related topic

FrobeniusMatrixNorm

Related topic

LpSpace

Related topic

CauchySchwarzInequality

Defines

Manhattan metric

Defines

Taxicab

Defines

L^1 norm

Defines

L^1 metric

Defines

L^2 metric

Defines

L^2 norm

Defines

L^∞norm

Generated on Fri Feb 9 21:36:47 2018 by

LaTeXML
l⁢i⁢m_{p→∞}ERROR(\delimiter)⁢69645069⁢x⁢ERROR(\delimiter)⁢86422285_p=ERROR(\delimiter)⁢69645069⁢x⁢ERROR(\delimiter)⁢86422285_∞

l

⁢

i

⁢

m

p

→

∞

\delimiter

⁢

69645069

⁢

x

⁢

\delimiter

⁢

86422285

p

=

\delimiter

⁢

69645069

⁢

x

⁢

\delimiter

⁢

86422285

∞

Suppose

x

=

(

x

1

,

…

,

x

n

)

is a point in

ℝ

n

, and let

∥

x

∥

p

and

∥

x

∥

∞

be the usual

p

-norm and

∞

-norm;

∥

x

∥

p

=

(

|

x

1

|

p

+

⋯

+

|

x

n

|

p

)

1

/

p

,

∥

x

∥

∞

=

max

⁡

{

|

x

1

|

,

…

,

|

x

n

|

}

.

Our claim is that

lim

p

→

∞

⁡

∥

x

∥

p

=

∥

x

∥

∞

.

(1)

In other words, for any fixed

x

∈

ℝ

n

, the above limit holds.
This, or course, justifies the notation for the

∞

-norm.

Proof.

Since both norms
stay

invariant

if we exchange two

components

in

x

, we can arrange things
such that

∥

x

∥

∞

=

|

x

1

|

. Then for any real

p

>

0

, we have

∥

x

∥

∞

=

|

x

1

|

=

(

|

x

1

|

p

)

1

/

p

≤

∥

x

∥

p

and

∥

x

∥

p

≤

n

1

/

p

⁢

|

x

1

|

=

n

1

/

p

⁢

∥

x

∥

∞

.

Taking the limit of the above inequalities (see
this page (

http://planetmath.org/InequalityForRealNumbers

))
we obtain

∥

x

∥

∞

≤

lim

p

→

∞

⁡

∥

x

∥

p

,

lim

p

→

∞

⁡

∥

x

∥

p

≤

∥

x

∥

∞

,

which combined yield the result.

□

Title

l

⁢

i

⁢

m

p

→

∞

\delimiter

⁢

69645069

⁢

x

⁢

\delimiter

⁢

86422285

p

=

\delimiter

⁢

69645069

⁢

x

⁢

\delimiter

⁢

86422285

∞

Canonical name

limptoinftylVertXrVertplVertXrVertinfty

Date of creation

2013-03-22 14:02:53

Last modified on

2013-03-22 14:02:53

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

12

Author

Koro (127)

Entry type

Result

Classification

msc 46B20

Related topic

PowerMean

Generated on Fri Feb 9 21:36:49 2018 by

LaTeXML
proof that L^p spaces are complete

proof that

L

p

spaces are complete

Let’s prove completeness for the classical

Banach spaces

, say

L

p

⁢

[

0

,

1

]

where

p

≥

1

.

Since the case

p

=

∞

is

elementary

, we may assume

1

≤

p

<

∞

.
Let

[

f

⋅

]

∈

(

L

p

)

𝐍

be a

Cauchy sequence

. Define

[

g

0

]

:=

[

f

0

]

and for

n

>

0

define

[

g

n

]

:=

[

f

n

-

f

n

-

1

]

. Then

[

∑

n

=

0

N

g

n

]

=

[

f

N

]

and we see that

∑

n

=

0

∞

∥

g

n

∥

=

∑

n

=

0

∞

∥

f

n

-

f

n

-

1

∥

≤

?

⁢

?

⁢

?

<

∞

.

Thus it suffices to prove that etc.

It suffices to prove that each absolutely summable series in

L

p

is
summable in

L

p

to some

element

in

L

p

.

Let

{

f

n

}

be a

sequence

in

L

p

with

∑

n

=

1

∞

∥

f

n

∥

=

M

<

∞

, and define

functions

g

n

by
setting

g

n

⁢

(

x

)

=

∑

k

=

1

n

|

f

k

⁢

(

x

)

|

. From the

Minkowski inequality

we
have

∥

g

n

∥

≤

∑

k

=

1

n

∥

f

k

∥

≤

M

.

Hence

∫

g

n

p

≤

M

p

.

For each

x

,

{

g

n

⁢

(

x

)

}

is an

increasing sequence

of (extended)

real
numbers

and so must

converge

to an

extended real number

g

⁢

(

x

)

. The
function

g

so defined is

measurable

, and, since

g

n

≥

0

, we have

∫

g

p

≤

M

p

by Fatou’s Lemma. Hence

g

p

is

integrable

, and

g

⁢

(

x

)

is finite for
almost all

x

.

For each

x

such that

g

⁢

(

x

)

is finite the series

∑

k

=

1

∞

f

k

⁢

(

x

)

is an absolutely summable series of real numbers and so must be summable
to a real number

s

⁢

(

x

)

. If we set

s

⁢

(

x

)

=

0

for those

x

where

g

⁢

(

x

)

=

∞

, we have defined a function

s

which is the limit almost
everywhere of the partial sums

s

n

=

∑

k

=

1

n

f

k

. Hence

s

is
measurable. Since

|

s

n

⁢

(

x

)

|

≤

g

⁢

(

x

)

, we have

|

s

⁢

(

x

)

|

≤

g

⁢

(

x

)

.
Consequently,

s

is in

L

p

and we have

|

s

n

⁢

(

x

)

-

s

⁢

(

x

)

|

p

≤

2

p

⁢

[

g

⁢

(

x

)

]

p

.

Since

2

p

⁢

g

p

is integrable and

|

s

n

⁢

(

x

)

-

s

⁢

(

x

)

|

p

converges to

0

for
almost all

x

, we have

∫

|

s

n

-

s

|

p

→

0

by the Lebesgue Convergence

Theorem

. Thus

∥

s

n

-

s

∥

p

→

0

, whence

∥

s

n

-

s

∥

→

0

. Consequently, the series

{

f

n

}

has in

L

p

the sum

s

.

References

Royden, H. L.

Real analysis

. Third edition

. Macmillan Publishing Company, New York, 1988.

Title

proof that

L

p

spaces are

complete

Canonical name

ProofThatLpSpacesAreComplete

Date of creation

2013-03-22 14:40:09

Last modified on

2013-03-22 14:40:09

Owner

Simone (5904)

Last modified by

Simone (5904)

Numerical id

8

Author

Simone (5904)

Entry type

Proof

Classification

msc 46B25

Generated on Fri Feb 9 21:36:52 2018 by

LaTeXML
all norms are not equivalent

all norms are not equivalent

Let

V

be the vector space of

continuous
functions

[

-

1

,

1

]

→

ℝ

that are

differentiable

at

0

.
Then we can define norms

∥

f

∥

=

max

x

∈

[

-

1

,

1

]

⁡

|

f

|

,

and

∥

f

∥

′

=

∥

f

∥

+

|

f

′

⁢

(

0

)

|

.

It is not difficult to find a

sequence

of

functions

f

1

,

f

2

,

…

in

V

such that

1.

f

k

′

⁢

(

0

)

=

k

for

k

=

1

,

2

,

…

,

2.

∥

f

k

∥

=

1

.

Then

∥

f

k

∥

=

1

, and

∥

f

k

∥

′

=

1

+

k

, so there is no

C

>

1

such that

∥

f

∥

′

≤

C

⁢

∥

f

∥

f

∈

V

,

and

∥

⋅

∥

and

∥

⋅

∥

′

cannot be .

Title

all norms are not

equivalent

Canonical name

AllNormsAreNotEquivalent

Date of creation

2013-03-22 15:36:11

Last modified on

2013-03-22 15:36:11

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Example

Classification

msc 46B99

Generated on Fri Feb 9 21:36:57 2018 by

LaTeXML
approximation property

approximation property

Let

Y

be a

Banach space

and

B

⁢

(

Y

)

the

algebra

of

bounded operators

in

Y

. We say that

Y

has the

approximation property

if there is a

sequence

(

P

n

)

of finite rank (

http://planetmath.org/RankLinearMapping

)

operators

in

B

⁢

(

Y

)

such that

P

n

⁢

y

⟶

y

⁢

∀

y

∈

Y

i.e.

(

P

n

)

converges

in the

strong operator topology

to the

identity operator

.

The fundamental fact about spaces with the approximation property is that every

compact operator

is the norm limit of finite rank operators.

Theorem

-

Let

X

be a

normed vector space

and

Y

a Banach space with the approximation property. Then every compact operator

T

:

X

⟶

Y

is the norm limit of operators of finite rank.

Examples :

•

Separable Hilbert spaces have the approximation property. Note however that compact operators on Hilbert spaces (not just

separable

ones) are always norm limit of finite rank operators.

•

The

ℓ

p

-spaces (

http://planetmath.org/Lp

) have the approximation property.

Moreover,

Theorem -

If

Y

is a Banach space with a

Schauder basis

then it has the approximation property.

Title

approximation property

Canonical name

ApproximationProperty

Date of creation

2013-03-22 17:28:42

Last modified on

2013-03-22 17:28:42

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46B99

Synonym

approximation by finite rank operators

Defines

Schauder basis and approximation by finite rank operators

Generated on Fri Feb 9 21:37:03 2018 by

LaTeXML
Banach space

Banach space

A

Banach space

(

X

,

∥

⋅

∥

)

is a

normed vector space

such that

X

is

complete

under the metric

induced

by the norm

∥

⋅

∥

.

Some authors use the term Banach space only in the case where

X

is

infinite-dimensional

, although on Planetmath finite-dimensional spaces are also considered to be Banach spaces.

If

Y

is a Banach space and

X

is any normed vector space, then the set of

continuous

linear maps

f

:

X

→

Y

forms a Banach space, with norm given by the

operator norm

. In particular, since

ℝ

and

ℂ

are complete, the continuous

linear functionals

on a normed vector space

ℬ

form a Banach space, known as the

dual space

of

ℬ

.

Examples:

•

Finite-dimensional normed vector spaces (

http://planetmath.org/EveryFiniteDimensionalNormedVectorSpaceIsABanachSpace

).

•

L

p

spaces (

http://planetmath.org/LpSpace

) are by far the most common example of Banach spaces.

•

ℓ

p

spaces (

http://planetmath.org/Lp

) are

L

p

spaces for the counting measure on

ℕ

.

•

Continuous functions

on a

compact set

under the

supremum norm

.

•

Finite (

http://planetmath.org/FiniteMeasureSpace

) signed measures on a

σ

-algebra (

http://planetmath.org/SigmaAlgebra

).

Title

Banach space

Canonical name

BanachSpace

Date of creation

2013-03-22 12:13:48

Last modified on

2013-03-22 12:13:48

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

11

Author

bbukh (348)

Entry type

Definition

Classification

msc 46B99

Classification

msc 54E50

Related topic

VectorNorm

Related topic

DualSpace

Defines

dual space

Generated on Fri Feb 9 21:37:04 2018 by

LaTeXML
Banach-Steinhaus theorem

Banach-Steinhaus theorem

Let

X

be a

Banach space

and

Y

a

normed space

.
If a family

ℱ

⊂

ℬ

⁢

(

X

,

Y

)

of

bounded operators

from

X

to

Y

satisfies

sup

⁡

{

∥

T

⁢

(

x

)

∥

:

T

∈

ℱ

}

<

∞

for each

x

∈

X

, then

sup

⁡

{

∥

T

∥

:

T

∈

ℱ

}

<

∞

,

i.e.

ℱ

is a

bounded

subset of

ℬ

⁢

(

X

,

Y

)

with the usual

operator norm

. In other words,
there exists a constant

c

such that for all

x

∈

X

and

T

∈

ℱ

,

∥

T

⁢

x

∥

≤

c

⁢

∥

x

∥

.

Title

Banach-Steinhaus theorem

Canonical name

BanachSteinhausTheorem

Date of creation

2013-03-22 14:48:39

Last modified on

2013-03-22 14:48:39

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Theorem

Classification

msc 46B99

Synonym

Principle of Uniform Boundedness

Synonym

Uniform Boundedness Principle

Generated on Fri Feb 9 21:37:06 2018 by

LaTeXML