aarjaneiro/planetmath huggingface dataset:
properties of the transpose operator

properties of the transpose operator

In this entry, whenever

V

,

W

are

normed vector spaces

,

ℬ

⁢

(

V

,

W

)

denotes the algebra of

bounded linear operators

V

⟶

W

.

Let

X

,

Y

,

Z

be normed vector spaces and

X

′

,

Y

′

,

Z

′

be their continuous dual spaces. Let

T

,

S

∈

ℬ

⁢

(

X

,

Y

)

,

R

∈

ℬ

⁢

(

Y

,

Z

)

and

λ

∈

ℂ

.

0.0.1

Basic properties

•

T

′

∈

ℬ

⁢

(

Y

′

,

X

′

)

and

∥

T

∥

=

∥

T

′

∥

.

•

(

λ

⁢

T

)

′

=

λ

⁢

T

′

.

•

(

S

+

T

)

′

=

S

′

+

T

′

.

•

(

R

⁢

T

)

′

=

T

′

⁢

R

′

.

•

If

T

-

1

exists and

T

-

1

∈

ℬ

⁢

(

Y

,

X

)

then

(

T

′

)

-

1

∈

ℬ

⁢

(

X

′

,

Y

′

)

and

(

T

′

)

-

1

=

(

T

-

1

)

′

.

0.0.2

Miscellaneous properties

•

If we endow

X

′

and

Y

′

with the weak-*

topology

then

T

′

:

Y

′

⟶

X

′

is

continuous

.

•

T

is an

isometric isomorphism

if and only if

T

′

is an isometric isomorphism.

•

If

T

is

compact

(

http://planetmath.org/CompactOperator

) then

T

′

is also compact.

•

If

T

′

is compact and

Y

is a

Banach space

, then

T

is also compact.

Title

properties of the

transpose operator

Canonical name

PropertiesOfTheTransposeOperator

Date of creation

2013-03-22 17:36:02

Last modified on

2013-03-22 17:36:02

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46-00

Classification

msc 47A05

Synonym

transpose operator properties

Generated on Fri Feb 9 21:34:42 2018 by

LaTeXML
sequential characterization of boundedness

sequential characterization of boundedness

Theorem

[

1

,

2

]

A set

B

in a real (or possibly complex)

topological vector space

V

is

bounded

(

http://planetmath.org/BoundedSetInATopologicalVectorSpace

) if and only if the
following condition holds:

If

{

z

i

}

i

=

1

∞

is a sequence in

B

, and

{

λ

i

}

i

=

1

∞

is a sequence of scalars (in

ℝ

or

ℂ

),
such that

λ

i

→

0

, then

λ

i

⁢

z

i

→

0

in

V

.

References

1

W. Rudin,

Functional Analysis

,
McGraw-Hill Book Company, 1973.

2

R. Cristescu,

Topological vector spaces

,
Noordhoff International Publishing, 1977.

Title

sequential characterization of boundedness

Canonical

name

SequentialCharacterizationOfBoundedness

Date of creation

2013-03-22 13:48:17

Last modified on

2013-03-22 13:48:17

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

8

Author

bwebste (988)

Entry type

Theorem

Classification

msc 46-00

Generated on Fri Feb 9 21:34:43 2018 by

LaTeXML
topology of locally convex spaces is generated by seminorms

topology of locally convex spaces is generated by seminorms

Theorem -

Let

V

be a

topological vector space

over

ℝ

or

ℂ

. Then

V

is locally convex (

http://planetmath.org/LocallyConvexTopologicalVectorSpace

) if and only if the

topology

of

V

is a family of

seminorms

.

Moreover,

V

is

Hausdorff

and locally if and only if the topology of

V

is a family of seminorms.

Title

topology of locally convex spaces is generated by seminorms

Canonical name

TopologyOfLocallyConvexSpacesIsGeneratedBySeminorms

Date of creation

2013-03-22 17:43:29

Last modified on

2013-03-22 17:43:29

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46-00

Classification

msc 46A03

Generated on Fri Feb 9 21:34:45 2018 by

LaTeXML
weak convergence

weak convergence

Suppose

X

is a

topological vector space

,

X

′

is the continuous dual
of

X

, and

x

0

,

x

1

,

…

is a

sequence

in

X

.
Then we say that

x

i

converges

weakly

to

x

∈

X

if

lim

i

→

∞

⁡

f

⁢

(

x

i

)

=

f

⁢

(

x

)

for every

f

∈

X

′

. The notation for this is

x

i

→

𝑤

x

.

Title

weak convergence

Canonical name

WeakConvergence

Date of creation

2013-03-22 15:00:58

Last modified on

2013-03-22 15:00:58

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

8

Author

matte (1858)

Entry type

Definition

Classification

msc 46-00

Related topic

WeakConvergenceInNormedLinearSpace

Related topic

ConvergenceInDistribution

Generated on Fri Feb 9 21:34:47 2018 by

LaTeXML
Krein-Milman theorem

Krein-Milman theorem

Theorem

.

Let

X

be a

locally convex topological vector space

, and let

K

⊂

X

be a

compact

convex subset (

http://planetmath.org/ConvexSet

). Then

K

is the closed

convex hull

of its

extreme points

.

The closed convex hull above is defined as the intersection of all closed convex subsets of

X

that contain

K

. This turns out to be the same as the closure of the convex hull in a

topological vector space

.

References

1

H. L. Royden.

. Prentice-Hall, Englewood Cliffs, New Jersey, 1988

Title

Krein-Milman theorem

Canonical name

KreinMilmanTheorem

Date of creation

2013-03-22 14:24:58

Last modified on

2013-03-22 14:24:58

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Theorem

Classification

msc 46A03

Classification

msc 52A07

Classification

msc 52A99

Generated on Fri Feb 9 21:34:54 2018 by

LaTeXML
locally convex topological vector space

locally convex topological vector space

Definition

Let

V

be a

topological vector space

over a

subfield

of the

complex numbers

(usually taken to be

ℝ

or

ℂ

). If the topology of

V

has a basis where each member is
a

convex set

, then

V

is a

locally convex topological
vector space

[

1

]

.

Though most

vector spaces

occurring in practice are locally convex, the spaces

L

p

for

0

<

p

<

1

are examples of spaces which are not locally convex.

References

1

G.B. Folland,

Real Analysis: Modern Techniques and Their Applications

, 2nd ed, John Wiley & Sons, Inc., 1999.

Title

locally convex topological vector space

Canonical name

LocallyConvexTopologicalVectorSpace

Date of creation

2013-03-22 13:44:03

Last modified on

2013-03-22 13:44:03

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 46A03

Classification

msc 46-00

Generated on Fri Feb 9 21:34:56 2018 by

LaTeXML
weak-* topology

weak-* topology

Let

X

be a

locally convex topological vector space

(over

ℂ

or

ℝ

), and let

X

*

be the set of

continuous

linear functionals

on

X

(the

continuous dual

of

X

).
If

f

∈

X

*

then let

p

f

denote the

seminorm

p

f

⁢

(

x

)

=

|

f

⁢

(

x

)

|

, and let

p

x

⁢

(

f

)

denote the seminorm

p

x

⁢

(

f

)

=

|

f

⁢

(

x

)

|

.
Obviously any

normed space

is a locally convex topological vector space so

X

could be a normed space.

Definition

.

The topology on

X

defined by the seminorms

{

p

f

∣

f

∈

X

*

}

is called the

weak topology

and the topology on

X

*

defined by the seminorms

{

p

x

∣

x

∈

X

}

is called the

weak-

*

topology

.

The weak topology on

X

is usually denoted by

σ

⁢

(

X

,

X

*

)

and the weak-

*

topology on

X

*

is usually denoted by

σ

⁢

(

X

*

,

X

)

. Another common notation is

(

X

,

w

⁢

k

)

and

(

X

*

,

w

k

-

*

)

Topology defined on a space

Y

by seminorms

p

ι

,

ι

∈

I

means that we take the sets

{

y

∈

Y

∣

p

ι

⁢

(

y

)

<

ϵ

}

for all

ι

∈

I

and

ϵ

>

0

as a subbase for the topology (that is finite

intersections

of such sets form the basis).

The most striking result about weak-

*

topology is the Alaoglu’s

theorem

which asserts that for

X

being a normed space, a

closed ball

(in the

operator norm

) of

X

*

is weak-

*

compact

. There is no

similar

result for the weak topology on

X

, unless

X

is a reflexive space.

Note that

X

*

is sometimes used for the algebraic dual of a space and

X

′

is used for the continuous dual. In

functional analysis

X

*

always means the continuous dual and hence the term

weak-

*

topology

.

References

1

John B. Conway.
,
Springer-Verlag, New York, New York, 1990.

Title

weak-* topology

Canonical name

WeakTopology

Date of creation

2013-03-22 15:07:05

Last modified on

2013-03-22 15:07:05

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Definition

Classification

msc 46A03

Synonym

weak-* topology

Synonym

weak-

*

topology

Synonym

weak-star topology

Related topic

WeakHomotopyAdditionLemma

Defines

weak topology

Generated on Fri Feb 9 21:35:01 2018 by

LaTeXML
weak-* topology of the space of Radon measures

weak-* topology of the space of Radon measures

Let

X

be a

locally compact Hausdorff space

.
Let

M

⁢

(

X

)

denote the space of complex

Radon measures

on

X

, and

C

0

⁢

(

X

)

*

denote the dual of the

C

0

⁢

(

X

)

, the complex-valued

continuous functions

on

X

vanishing at infinity, equipped with the

uniform norm

.
By the

Riesz Representation Theorem

,

M

⁢

(

X

)

is

isometric

to

C

0

⁢

(

X

)

*

,
The isometry maps a

measure

μ

into the

linear functional

I

μ

⁢

(

f

)

=

∫

X

f

⁢

𝑑

μ

.

The weak-*

topology

(also called the vague topology) on

C

0

⁢

(

X

)

*

,
is simply the topology of

pointwise convergence

of

I

μ

:

I

μ

α

→

I

μ

if and only if

I

μ

α

⁢

(

f

)

→

I

μ

⁢

(

f

)

for each

f

∈

C

0

⁢

(

X

)

.

The corresponding topology on

M

⁢

(

X

)

induced by the isometry from

C

0

⁢

(

X

)

*

is also called
the weak-* or vague topology on

M

⁢

(

X

)

. Thus one may talk about “

weak convergence

” of measures

μ

n

→

μ

. One of the most important applications of this notion is in probability theory:
for example, the

central limit theorem

is essentially the statement that
if

μ

n

are the

distributions

for certain sums of

independent

random variables

.
then

μ

n

converge

weakly to a

normal distribution

,
i.e. the distribution

μ

n

is “approximately normal” for large

n

.

References

1

G.B. Folland,

Real Analysis

: Modern Techniques and Their Applications

, 2nd ed, John Wiley & Sons, Inc., 1999.

Title

weak-* topology of the space of Radon measures

Canonical name

WeakTopologyOfTheSpaceOfRadonMeasures

Date of creation

2013-03-22 15:22:58

Last modified on

2013-03-22 15:22:58

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

4

Author

stevecheng (10074)

Entry type

Example

Classification

msc 46A03

Classification

msc 28A33

Related topic

LocallyCompactHausdorffSpace

Generated on Fri Feb 9 21:35:02 2018 by

LaTeXML
absorbing set

absorbing set

Let

V

be a

vector space

over a field

F

equipped with a
non-discrete

valuation

|

⋅

|

:

F

→

ℝ

. Let

A

,

B

be
two subsets of

V

. Then

A

is said to

absorb

B

if there
is a non-negative real number

r

such that, for all

λ

∈

F

with

|

λ

|

≥

r

,

B

⊆

λ

⁢

A

.

A

is said to
be an

absorbing set

, or a

radial subset

of

V

if

A

absorbs all finite subsets of

V

.

Equivalently,

A

is

absorbing

if for any

x

∈

V

, there is a
non-negative real number

r

such that

x

∈

λ

⁢

A

for all

λ

∈

F

with

|

λ

|

≥

r

. If a finite subset

B

of

V

consists of

x

1

,

…

,

x

n

, then corresponding to each

x

i

, there is an

r

i

≥

0

such that

x

i

∈

λ

⁢

A

such that

|

λ

∣

≥

r

i

,

∀

λ

∈

F

. So

x

i

∈

λ

⁢

A

with

|

λ

|

≥

r

if we take

r

=

max

⁡

{

r

1

,

…

,

r

n

}

. So

A

absorbs

B

.

Example

. If

V

=

ℝ

n

and

F

=

ℝ

, then any
set containing an open ball centered at

0

is absorbing. This
implies that an absorbing set does not have to be

connected

, convex.

A closely related concept is that of a

circled set

, or a

balanced set

. Let

V

and

F

be defined as above. A subset

C

of

V

is said to be

circled

, or

balanced

, if

λ

⁢

C

⊆

C

for all

|

λ

|

≤

1

. Clearly,

C

absorbs itself (

C

⊆

λ

-

1

⁢

C

,

|

λ

-

1

|

≥

1

), and

0

∈

C

.

C

is also

symmetric

(

-

C

=

C

), for

-

C

⊆

C

and

C

=

-

(

-

C

)

⊆

-

C

. As an
example of a circled set that is neither absorbing nor convex,
consider

V

=

ℝ

2

and

F

=

ℝ

, and

C

the union of

x

and

y

axes. For an example of an absorbing set that is not
circled, take the union of a unit disk and an annulus centered at 0
that is large enough so it is disjoint from the disk.

Title

absorbing set

Canonical name

AbsorbingSet

Date of creation

2013-03-22 15:26:24

Last modified on

2013-03-22 15:26:24

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 46A08

Classification

msc 15A03

Related topic

BalancedSet

Related topic

AbsorbingElement

Defines

absorbing

Defines

absorb

Defines

radial

Generated on Fri Feb 9 21:35:05 2018 by

LaTeXML
barrel

barrel

Let

V

be a

topological vector space

(TVS). A

barrel

B

is a subset of

V

that is closed, convex, balanced (

http://planetmath.org/BalancedSet

), and

absorbing

. For example, in a

Banach space

A

, any ball

{

v

∈

A

∣

||

v

||

≤

r

}

for some

r

>

0

is a barrel.

A topological vector space is said to be a

barrelled space

if it is locally convex (

http://planetmath.org/LocallyConvexTopologicalVectorSpace

), and every barrel is a

neighborhood

of

0

. Every Banach space is a barrelled space.

A weaker form of a barrelled space is that of an

infrabarrelled space

. A TVS is said to be

infrabarrelled

if it is locally convex, and every barrel that absorbs every

bounded set

is a neighborhood of

0

.

Let

V

be a

vector space

and

𝔗

be the set of all those

topologies

on

V

making

V

a TVS. In other words, if

T

∈

𝔗

, then

(

V

,

T

)

is a topological vector space.

Let

V

and

T

∈

𝔗

be defined as above. Then

(

V

,

T

)

being barrelled has an

equivalent

characterization

below:

(*) for any

T

1

∈

𝔗

such that there is a

neighborhood base

(

http://planetmath.org/LocalBase

) of

0

consisting of

T

-

closed sets

, then

T

1

is

coarser

than

T

.

A variation of a barrelled space is that of an

ultrabarrelled space

. A topological vector space is said to be

ultrabarrelled

if it

satisfies

(*) above. A locally convex ultrabarrelled space is barrelled.

References

1

H. H. Schaefer,

Topological Vector Spaces

, Springer-Verlag, New York (1970).

2

R. E. Edwards,

Functional Analysis

, Theory and Applications

, Holt, Reinhart and Winston, New York (1965).

Title

barrel

Canonical name

Barrel

Date of creation

2013-03-22 16:41:26

Last modified on

2013-03-22 16:41:26

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 46A08

Synonym

barreled space

Synonym

infrabarreled space

Synonym

ultrabarreled space

Synonym

barrelled

Synonym

infrabarrelled

Synonym

ultrabarrelled

Defines

barrelled space

Defines

infrabarrelled space

Defines

ultrabarrelled space

Generated on Fri Feb 9 21:35:14 2018 by

LaTeXML
bornological space

bornological space

A

bornivore

is a set which absorbs all

bounded sets

.
That is,

G

is a bornivore if given any bounded set

B

, there exists a

δ

>

0

such that

ϵ

⁢

B

⊂

G

for

0

≤

ϵ

<

δ

.

A

locally convex topological vector space

is said to be

bornological

if every convex bornivore is a

neighborhood

of 0.

A

metrizable

topological vector space

is bornological.

References

1

A. Wilansky,

Functional Analysis

, Blaisdell Publishing Co. 1964.

2

H.H. Schaefer, M. P. Wolff,

Topological Vector Spaces

,
2nd ed. 1999, Springer-Verlag.

Title

bornological space

Canonical

name

BornologicalSpace

Date of creation

2013-03-22 15:59:09

Last modified on

2013-03-22 15:59:09

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 46A08

Defines

bornivore

Generated on Fri Feb 9 21:35:15 2018 by

LaTeXML
bounded set

bounded set

A set in a

topological vector space

is

bounded

if it is absorbed by every

neighborhood

of 0. That is,

B

is bounded if for every neighborhood

U

of 0, there
is a

δ

>

0

such that

ϵ

⁢

B

⊂

U

for

0

≤

ϵ

<

δ

.

Title

bounded set

Canonical name

BoundedSet

Date of creation

2013-03-22 15:59:12

Last modified on

2013-03-22 15:59:12

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

4

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 46A08

Generated on Fri Feb 9 21:35:17 2018 by

LaTeXML
closed graph theorem

closed graph theorem

A linear mapping between two

Banach spaces

X

and

Y

is

continuous

if and only if its graph (

http://planetmath.org/Graph2

) is a

closed subset

of

X

×

Y

(with the

product topology

).

Title

closed graph theorem

Canonical name

ClosedGraphTheorem

Date of creation

2013-03-22 13:13:56

Last modified on

2013-03-22 13:13:56

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Theorem

Classification

msc 46A30

Related topic

OpenMappingTheorem

Related topic

Graph2

Generated on Fri Feb 9 21:35:19 2018 by

LaTeXML
proof of closed graph theorem

proof of closed graph theorem

Let

T

:

X

→

Y

be a linear mapping. Denote its graph by

G

⁢

(

T

)

, and let

p

1

:

X

×

Y

→

X

and

p

2

:

X

×

Y

→

Y

be the projections onto

X

and

Y

, respectively. We remark that these projections are

continuous

, by definition of the

product

of

Banach spaces

.

If

T

is

bounded

, then given a

sequence

{

(

x

i

,

T

⁢

x

i

)

}

in

G

⁢

(

T

)

which

converges

to

(

x

,

y

)

∈

X

×

Y

, we have that

x

i

=

p

1

⁢

(

x

i

,

T

⁢

x

i

)

→

i

→

∞

p

1

⁢

(

x

,

y

)

=

x

and

T

⁢

x

i

=

p

2

⁢

(

x

i

,

T

⁢

x

i

)

→

i

→

∞

p

2

⁢

(

x

,

y

)

=

y

,

by

continuity

of the projections.
But then, since

T

is continuous,

T

⁢

x

=

lim

i

→

∞

⁡

T

⁢

x

i

=

y

.

Thus

(

x

,

y

)

=

(

x

,

T

⁢

x

)

∈

G

⁢

(

T

)

, proving that

G

⁢

(

T

)

is closed.

Now suppose

G

⁢

(

T

)

is closed. We remark that

G

⁢

(

T

)

is a vector subspace of

X

×

Y

, and being closed, it is a Banach space. Consider the

operator

T

~

:

X

→

G

⁢

(

T

)

defined by

T

~

⁢

x

=

(

x

,

T

⁢

x

)

. It is clear that

T

~

is a

bijection

, its

inverse

being

p

1

|

G

⁢

(

T

)

, the

restriction

of

p

1

to

G

⁢

(

T

)

. Since

p

1

is continuous on

X

×

Y

, the restriction is continuous as well; and since it is also

surjective

, the

open mapping theorem

implies that

p

1

|

G

⁢

(

T

)

is an

open mapping

, so its inverse must be continuous. That is,

T

~

is continuous, and consequently

T

=

p

2

∘

T

~

is continuous.

Title

proof of closed graph theorem

Canonical name

ProofOfClosedGraphTheorem

Date of creation

2013-03-22 14:48:47

Last modified on

2013-03-22 14:48:47

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Proof

Classification

msc 46A30

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

LaTeXML
absolute value in a vector lattice

absolute value in a vector lattice

Let

V

be a

vector lattice

over

ℝ

, and

V

+

be its

positive cone

. We define three functions from

V

to

V

+

as follows. For any

x

∈

V

,

•

x

+

:=

x

∨

0

,

•

x

-

:=

(

-

x

)

∨

0

,

•

|

x

|

:=

(

-

x

)

∨

x

.

It is easy to see that these functions are well-defined. Below are some properties of the three functions:

1.

x

+

=

(

-

x

)

-

and

x

-

=

(

-

x

)

+

.

2.

x

=

x

+

-

x

-

, since

x

+

-

x

-

=

(

x

∨

0

)

-

(

-

x

)

∨

0

=

(

x

∨

0

)

+

(

x

∧

0

)

=

x

+

0

=

x

.

3.

|

x

|

=

x

+

+

x

-

, since

x

+

+

x

-

=

x

+

2

⁢

x

-

=

x

+

(

-

2

⁢

x

)

∨

0

=

(

x

-

2

⁢

x

)

∨

(

x

+

0

)

=

|

x

|

.

4.

If

0

≤

x

, then

x

+

=

x

,

x

-

=

0

and

|

x

|

=

x

. Also,

x

≤

0

implies

x

+

=

0

,

x

-

=

-

x

and

|

x

|

=

-

x

.

5.

|

x

|

=

0

iff

x

=

0

. The “only if” part is obvious. For the “if” part, if

|

x

|

=

0

, then

(

-

x

)

∨

x

=

0

, so

x

≤

0

and

-

x

≤

0

. But then

0

≤

x

, so

x

=

0

.

6.

|

r

⁢

x

|

=

|

r

|

⁢

|

x

|

for any

r

∈

ℝ

. If

0

≤

r

, then

|

r

⁢

x

|

=

(

-

r

⁢

x

)

∨

(

r

⁢

x

)

=

r

⁢

(

(

-

x

)

∨

x

)

=

r

⁢

|

x

|

=

|

r

|

⁢

|

x

|

. On the other hand, if

r

≤

0

, then

|

r

⁢

x

|

=

(

-

r

⁢

x

)

∨

(

r

⁢

x

)

=

(

-

r

)

⁢

(

x

∨

(

-

x

)

)

=

-

r

⁢

|

x

|

=

|

r

|

⁢

|

x

|

.

7.

|

x

|

+

|

y

|

=

|

x

+

y

|

∨

|

x

-

y

|

, since

L

⁢

H

⁢

S

=

(

-

x

)

∨

x

+

(

-

y

)

∨

y

=

(

-

x

-

y

)

∨

(

-

x

+

y

)

∨

(

x

-

y

)

∨

(

x

+

y

)

=

R

⁢

H

⁢

S

.

8.

(

triangle inequality

).

|

x

+

y

|

≤

|

x

|

+

|

y

|

, since

|

x

+

y

|

≤

|

x

+

y

|

∨

|

x

-

y

|

=

|

x

|

+

|

y

|

.

Properties 5, 6, and 8 satisfy the axioms of an

absolute value

, and therefore

|

x

|

is called the

absolute value

of

x

. However, it is not the “norm” of a vector in the traditional sense, since it is not a real-valued function.

Title

absolute value in a vector lattice

Canonical name

AbsoluteValueInAVectorLattice

Date of creation

2013-03-22 17:03:16

Last modified on

2013-03-22 17:03:16

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 46A40

Classification

msc 06F20

Defines

absolute value

Generated on Fri Feb 9 21:35:30 2018 by

LaTeXML
conical neighborhood

conical neighborhood

Let

V

be a real (or complex)

topological vector space

and

X

a subset of

V

.
A

conical neighborhood

of

X

is a cone in

V

which is a

neighborhood

of

X

.

In the theory of

pseudodifferential operators

and in microlocal analysis, we have to consider conical neighborhood of vectors located in some fiber of a vector bundle.

Title

conical neighborhood

Canonical

name

ConicalNeighborhood

Date of creation

2013-03-22 15:19:33

Last modified on

2013-03-22 15:19:33

Owner

urelement (1799)

Last modified by

urelement (1799)

Numerical id

7

Author

urelement (1799)

Entry type

Definition

Classification

msc 46A55

Synonym

conic neighborhood

Defines

conical neighborhood

Generated on Fri Feb 9 21:35:44 2018 by

LaTeXML
Heine-Cantor theorem

Heine-Cantor theorem

Let

X

,

Y

be

uniform spaces

, and

f

:

X

→

Y

a

continuous function

. If

X

is

compact

, then

f

is

uniformly continuous

.

For

instance

, if

f

:

[

a

,

b

]

→

ℝ

is a continuous function, then it is uniformly continuous.

Title

Heine-Cantor theorem

Canonical name

HeineCantorTheorem

Date of creation

2013-03-22 12:45:32

Last modified on

2013-03-22 12:45:32

Owner

n3o (216)

Last modified by

n3o (216)

Numerical id

9

Author

n3o (216)

Entry type

Theorem

Classification

msc 46A99

Generated on Fri Feb 9 21:35:51 2018 by

LaTeXML