aarjaneiro/planetmath huggingface dataset:
orthogonal decomposition theorem

orthogonal decomposition theorem

Theorem

-

Let

X

be an

Hilbert space

and

A

⊆

X

a closed subspace. Then the

orthogonal complement

(

http://planetmath.org/Complimentary

) of

A

, denoted

A

⟂

, is a

topological complement

of

A

. That means

A

⟂

is closed and

X

=

A

⊕

A

⟂

.

Proof :

•

A

⟂

is closed :

This follows easily from the continuity of the

inner product

. If a

sequence

(

x

n

)

of

elements

in

A

⟂

converges

to an element

x

0

∈

X

, then

⟨

x

0

,

a

⟩

=

⟨

lim

n

→

∞

⁡

x

n

,

a

⟩

=

lim

n

→

∞

⁡

⟨

x

n

,

a

⟩

=

0

⁢

for every

⁢

a

∈

A

which implies that

x

0

∈

A

⟂

.

•

X

=

A

⊕

A

⟂

:

Since

X

is

complete

(

http://planetmath.org/Complete

) and

A

is closed,

A

is a

subspace

of

X

. Therefore, for every

x

∈

X

, there exists a best approximation of

x

in

A

, which we denote by

a

0

∈

A

, that

satisfies

x

-

a

0

∈

A

⟂

(see this entry (

http://planetmath.org/BestApproximationInInnerProductSpaces

)).

This allows one to write

x

as a sum of elements in

A

and

A

⟂

x

=

a

0

+

(

x

-

a

0

)

which proves that

X

=

A

+

A

⟂

.

Moreover, it is easy to see that

A

∩

A

⟂

=

{

0

}

since if

y

∈

A

∩

A

⟂

then

⟨

y

,

y

⟩

=

0

, which means

y

=

0

.

We conclude that

X

=

A

⊕

A

⟂

.

□

Title

orthogonal decomposition theorem

Canonical name

OrthogonalDecompositionTheorem

Date of creation

2013-03-22 17:32:34

Last modified on

2013-03-22 17:32:34

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46A99

Synonym

closed subspaces of Hilbert spaces are complemented

Generated on Fri Feb 9 21:35:54 2018 by

LaTeXML
proof of Heine-Cantor theorem

proof of Heine-Cantor theorem

We seek to show that

f

:

K

→

X

is

continuous

with

K

a

compact

metric space

, then

f

is

uniformly continuous

. Recall that for

f

:

K

→

X

, uniform continuity
is the condition that for any

ε

>

0

, there exists

δ

such that

d

K

⁢

(

x

,

y

)

<

δ

⟹

d

X

⁢

(

f

⁢

(

x

)

,

f

⁢

(

y

)

)

<

ϵ

for all

x

,

y

∈

K

Suppose

K

is a compact metric space,

f

continuous on

K

. Let

ϵ

>

0

. For each

k

∈

K

choose

δ

k

such that

d

⁢

(

k

,

x

)

≤

δ

k

implies

d

⁢

(

f

⁢

(

k

)

,

f

⁢

(

x

)

)

≤

ϵ

2

. Note that the collection of balls

B

⁢

(

k

,

δ

k

2

)

covers

K

, so by compactness there is a finite

subcover

,
say involving

k

1

,

…

,

k

n

. Take

δ

=

min

i

=

1

,

…

,

n

⁡

δ

k

i

2

Then, suppose

d

⁢

(

x

,

y

)

≤

δ

. By the choice of

k

1

,

…

,

k

n

and the

triangle inequality

, there exists an

i

such that

d

⁢

(

x

,

k

i

)

,

d

⁢

(

y

,

k

i

)

≤

δ

k

i

. Hence,

d

⁢

(

f

⁢

(

x

)

,

f

⁢

(

y

)

)

≤

d

⁢

(

f

⁢

(

x

)

,

f

⁢

(

k

i

)

)

+

d

⁢

(

f

⁢

(

y

)

,

f

⁢

(

k

i

)

)

(1)

≤

ϵ

2

+

ϵ

2

(2)

As

x

,

y

were arbitrary, we have that

f

is uniformly continuous.

This proof is similar to one found in Mathematical Principles of

Analysis

, Rudin.

Title

proof of Heine-Cantor theorem

Canonical name

ProofOfHeineCantorTheorem

Date of creation

2013-03-22 15:09:43

Last modified on

2013-03-22 15:09:43

Owner

drini (3)

Last modified by

drini (3)

Numerical id

10

Author

drini (3)

Entry type

Proof

Classification

msc 46A99

Generated on Fri Feb 9 21:35:57 2018 by

LaTeXML
proof of Heine-Cantor theorem

proof of Heine-Cantor theorem

We prove this

theorem

in the case when

X

and

Y

are

metric spaces

.

Suppose

f

is not

uniformly continuous

. Then

∃

ϵ

>

0

⁢

∀

δ

>

0

⁢

∃

x

,

y

∈

X

d

⁢

(

x

,

y

)

<

δ

⁢

but

⁢

d

⁢

(

f

⁢

(

x

)

,

f

⁢

(

y

)

)

≥

ϵ

.

In particular by letting

δ

=

1

/

k

we can construct two

sequences

x

k

and

y

k

such that

d

(

x

k

,

y

k

)

<

1

/

k

and

d

(

f

(

x

k

)

,

f

(

y

k

)

≥

ϵ

.

Since

X

is

compact

the two sequence have convergent subsequences i.e.

x

k

j

→

x

¯

∈

X

,

y

k

j

→

y

¯

∈

X

.

Since

d

⁢

(

x

k

,

y

k

)

→

0

we have

x

¯

=

y

¯

. Being

f

continuous

we hence conclude

d

⁢

(

f

⁢

(

x

k

j

)

,

f

⁢

(

y

k

j

)

)

→

0

which is a

contradiction

being

d

⁢

(

f

⁢

(

x

k

)

,

f

⁢

(

y

k

)

)

≥

ϵ

.

Title

proof of Heine-Cantor theorem

Canonical name

ProofOfHeineCantorTheorem

Date of creation

2013-03-22 13:31:26

Last modified on

2013-03-22 13:31:26

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

5

Author

paolini (1187)

Entry type

Proof

Classification

msc 46A99

Generated on Fri Feb 9 21:36:00 2018 by

LaTeXML
proper subspaces of a topological vector space have empty interior

proper subspaces of a topological vector space have empty interior

Theorem -

Let

V

be a

topological vector space

. Every proper

subspace

S

⊂

V

has empty

interior

.

Proof :

Let

S

be a subspace of

V

. Suppose there is a non-empty

open set

A

⊆

S

.

Fix a point

a

0

∈

A

⊆

S

. Since the

vector sum

operation is

continuous

,

translations

of open sets are again open sets. In particular, the set

A

-

a

0

:=

{

x

-

a

0

:

x

∈

A

}

is an open set of

V

that contains the origin

0

.

As

S

is a vector subspace and

A

⊆

S

, we see that the translation

A

-

a

0

is still contained in

S

.

Since the

scalar multiplication

operation is continuous it follows easily that, for every

x

∈

V

, the function

f

x

:

𝕂

⟶

V

given by

f

x

⁢

(

λ

)

=

λ

⁢

x

is also continuous.

Consider now any vector

v

∈

V

. The set

f

v

-

1

⁢

(

A

-

a

0

)

is an open set that contains

0

. Thus, taking a value

λ

∈

f

v

-

1

⁢

(

A

-

a

0

)

we see that

λ

⁢

v

∈

A

-

a

0

,

i.e. we can multiply

v

by a sufficiently small

λ

such that

λ

⁢

v

belongs to the open set

A

-

a

0

.

Since the set

A

-

a

0

is contained in

S

, we see that

λ

⁢

v

∈

S

, and therefore

v

∈

S

.

This proves that

V

=

S

, i.e.

S

is not proper.

We conclude that if

S

is proper then

S

has empty interior.

□

Title

proper subspaces of a topological vector space have empty interior

Canonical name

ProperSubspacesOfATopologicalVectorSpaceHaveEmptyInterior

Date of creation

2013-03-22 17:34:23

Last modified on

2013-03-22 17:34:23

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46A99

Generated on Fri Feb 9 21:36:02 2018 by

LaTeXML
topological complement

topological complement

0.0.1

Definition

Let

X

be a

topological vector space

and

M

⊆

X

a closed (

http://planetmath.org/ClosedSet

)

subspace

.

If there exists a closed subspace

N

⊆

X

such that

M

⊕

N

=

X

we say that

M

is

topologically complemented

.

In this case

N

is said to be a

topological complement

of

M

, and also

M

and

N

are said to be

topologically complementary

subspaces.

0.0.2

Remarks

•

It is known that every subspace

M

⊆

X

has an

algebraic complement

, i.e. there exists a subspace

N

⊆

X

such that

M

⊕

N

=

X

. The

existence

of topological complements, however, is not always assured.

•

If

X

is an

Hilbert space

, then each closed subspace

M

⊆

X

is topologically complemented by its

orthogonal complement

M

⟂

, i.e.

M

⊕

M

⟂

=

X

.

•

Moreover, for

Banach spaces

the

converse

of the last paragraph also holds, i.e. if each closed subspace is topologically complemented then

X

is

isomorphic

a Hilbert space. This is the

Lindenstrauss-Tzafriri theorem

(

http://planetmath.org/CharacterizationOfAHilbertSpace

).

Title

topological complement

Canonical name

TopologicalComplement

Date of creation

2013-03-22 17:32:31

Last modified on

2013-03-22 17:32:31

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46A99

Classification

msc 15A03

Defines

topologically complementary

Defines

topologically complemented

Generated on Fri Feb 9 21:36:04 2018 by

LaTeXML
topological vector space

topological vector space

Definition

A

topological vector space

is a pair

(

V

,

𝒯

)

,
where

V

is a

vector space

over a

topological field

K

,
and

𝒯

is a

topology

on

V

such that under

𝒯

the

scalar multiplication

(

λ

,

v

)

↦

λ

⁢

v

is a

continuous function

K

×

V

→

V

and the

vector addition

(

v

,

w

)

↦

v

+

w

is a continuous function

V

×

V

→

V

,
where

K

×

V

and

V

×

V

are given the respective

product topologies

.

We will also require that

{

0

}

is closed
(which is

equivalent

to requiring the topology to be

Hausdorff

),
though some authors do not make this requirement.
Many authors require that

K

be either

ℝ

or

ℂ

(with their

usual topologies

).

Topological vector spaces as topological groups

A topological vector space is necessarily a topological group:
the definition ensures that the

group operation

(vector addition) is continuous,
and the

inverse

operation

is the same as

multiplication

by

-

1

,
and so is also continuous.

Finite-dimensional topological vector spaces

A

finite-dimensional

vector space inherits a natural topology.
For if

V

is a finite-dimensional vector space,
then

V

is

isomorphic

to

K

n

for some

n

;
then let

f

:

V

→

K

n

be such an

isomorphism

,
and suppose that

K

n

has the product topology.
Give

V

the topology where a subset

A

of

V

is open in

V

if and only if

f

⁢

(

A

)

is open in

K

n

.
This topology is independent of the choice of isomorphism

f

,
and is the finest (

http://planetmath.org/

Coarser

) topology on

V

that makes it into a topological vector space.

Title

topological vector space

Canonical name

TopologicalVectorSpace

Date of creation

2013-03-22 12:16:55

Last modified on

2013-03-22 12:16:55

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

22

Author

yark (2760)

Entry type

Definition

Classification

msc 46A99

Synonym

TVS

Synonym

linear topological space

Synonym

topological linear space

Related topic

TopologicalRing

Related topic

FrechetSpace

Generated on Fri Feb 9 21:36:06 2018 by

LaTeXML
Mazur-Ulam theorem

Mazur-Ulam theorem

Theorem

.

Every

isometry

(

http://planetmath.org/Isometry

) between

normed vector spaces

over

R

is an

affine transformation

.

Note that we consider isometries to be surjective by definition.
The result is not in general true for non-surjective isometric mappings.

The result does not extend to normed vector spaces over

ℂ

,
as can be seen from the fact that complex conjugation is an isometry

ℂ

→

ℂ

but is not affine over

ℂ

.
(But complex conjugation is clearly affine over

ℝ

,
and in general any normed vector space over

ℂ

can be considered as a normed vector space over

ℝ

,
to which the theorem can be applied.)

This theorem was first proved by Mazur and Ulam.

[

1

]

A simpler proof has been given by Jussi Väisälä.

[

2

]

References

1

S. Mazur and S. Ulam,

Sur les

transformations

isométriques d’espaces vectoriels normés

,
C. R. Acad. Sci., Paris 194 (1932), 946–948.

2

Jussi Väisälä,

A proof of the Mazur–Ulam theorem

,
Amer. Math. Mon. 110, #7 (2003), 633–635.
(A preprint is

http://www.helsinki.fi/%7Ejvaisala/mazurulam.pdf

available on Väisälä’s website.)

Title

Mazur-Ulam theorem

Canonical name

MazurUlamTheorem

Date of creation

2013-03-22 16:22:50

Last modified on

2013-03-22 16:22:50

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Theorem

Classification

msc 46B04

Generated on Fri Feb 9 21:36:08 2018 by

LaTeXML
Lipschitz inverse mapping theorem

Lipschitz inverse mapping theorem

Let

(

E

,

∥

⋅

∥

)

be a

Banach space

and let

A

:

E

→

E

be a

bounded

linear isomorphism

with
bounded

inverse

(i.e. a topological linear

automorphism

);
let

B

⁢

(

r

)

be the ball with center 0
and radius

r

(we allow

r

=

∞

). Then for any Lipschitz map

ϕ

:

B

⁢

(

r

)

→

E

such that

Lip

⁡

ϕ

<

∥

A

-

1

∥

-

1

and

ϕ

⁢

(

0

)

=

0

, there are open sets

U

⊂

E

and

V

⊂

B

⁢

(

r

)

and a map

T

:

U

→

V

such that

T

⁢

(

A

+

ϕ

)

=

I

|

V

and

(

A

+

ϕ

)

⁢

T

=

I

|

U

.
In other words, there is a local inverse of

A

+

ϕ

near zero. Furthermore, the inverse

T

is Lipschitz with

Lip

⁡

T

≤

(

∥

A

∥

+

Lip

⁡

ϕ

)

-

1

and

B

⁢

(

r

⁢

(

∥

A

-

1

∥

-

1

-

Lip

⁡

ϕ

)

)

⊂

U

.

Remark.

The inclusion above implies that

A

+

ϕ

:

E

→

E

is

invertible

if

r

=

∞

.

Remark.

Lip

⁡

ϕ

denotes the smallest Lipschitz constant of

ϕ

.

Title

Lipschitz inverse mapping theorem

Canonical name

LipschitzInverseMappingTheorem

Date of creation

2013-03-22 14:25:13

Last modified on

2013-03-22 14:25:13

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Theorem

Classification

msc 46B07

Classification

msc 47J07

Generated on Fri Feb 9 21:36:10 2018 by

LaTeXML
Banach-Alaoglu theorem

Banach-Alaoglu theorem

Let

X

be a

normed space

, and let

X

*

be its dual. Then the closed

unit ball

of

X

*

,

{

f

∈

X

*

:

∥

f

∥

≤

1

}

is

compact

in the weak-

*

topology

.

Title

Banach-Alaoglu theorem

Canonical name

BanachAlaogluTheorem

Date of creation

2013-03-22 14:48:44

Last modified on

2013-03-22 14:48:44

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Theorem

Classification

msc 46B10

Synonym

Alaoglu’s theorem

Generated on Fri Feb 9 21:36:11 2018 by

LaTeXML
corollary of Banach-Alaoglu theorem

corollary of Banach-Alaoglu theorem

Corollary

.

A

Banach space

H

is

isometrically isomorphic

to a closed subspace of

C

⁢

(

X

)

for a

compact

Hausdorff space

X

.

Proof.

Let

X

be the

unit ball

ℬ

⁢

(

ℋ

*

)

of

ℋ

*

. By the

Banach-Alaoglu theorem

it is compact in the weak-

*

topology

. Define the map

Φ

:

ℋ

→

C

⁢

(

X

)

by

(

Φ

⁢

f

)

⁢

(

φ

)

=

φ

⁢

(

f

)

. This is linear and we have for

f

∈

ℋ

:

∥

Φ

⁢

(

f

)

∥

∞

=

sup

φ

∈

ℬ

⁢

(

ℋ

*

)

⁡

|

Φ

⁢

(

f

)

⁢

(

φ

)

|

=

sup

φ

∈

ℬ

⁢

(

ℋ

*

)

⁡

|

φ

⁢

(

f

)

|

≤

sup

φ

∈

ℬ

⁢

(

ℋ

*

)

⁡

∥

φ

∥

⁢

∥

f

∥

≤

∥

f

∥

With the

Hahn-Banach theorem

it follows that there is a

φ

∈

ℬ

⁢

(

ℋ

*

)

such that

φ

⁢

(

f

)

=

∥

f

∥

. Thus

∥

Φ

⁢

(

f

)

∥

∞

=

∥

f

∥

and

Φ

is an isometric isomorphism, as required.
∎

Title

corollary of Banach-Alaoglu theorem

Canonical name

CorollaryOfBanachAlaogluTheorem

Date of creation

2013-03-22 18:34:45

Last modified on

2013-03-22 18:34:45

Owner

karstenb (16623)

Last modified by

karstenb (16623)

Numerical id

4

Author

karstenb (16623)

Entry type

Corollary

Classification

msc 46B10

Generated on Fri Feb 9 21:36:14 2018 by

LaTeXML
proof of Banach-Alaoglu theorem

proof of Banach-Alaoglu theorem

For any

x

∈

X

, let

D

x

=

{

z

∈

ℂ

:

|

z

|

≤

∥

x

∥

}

and

D

=

Π

x

∈

X

⁢

D

x

. Since

D

x

is a

compact subset

of

ℂ

,

D

is

compact

in

product topology

by

Tychonoff theorem

.

We prove the

theorem

by finding a

homeomorphism

that maps the closed

unit ball

B

X

*

of

X

*

onto a

closed subset

of

D

. Define

Φ

x

:

B

X

*

→

D

x

by

Φ

x

⁢

(

f

)

=

f

⁢

(

x

)

and

Φ

:

B

X

*

→

D

by

Φ

=

Π

x

∈

X

⁢

Φ

x

, so that

Φ

⁢

(

f

)

=

(

f

⁢

(

x

)

)

x

∈

X

. Obviously,

Φ

is

one-to-one

, and a net

(

f

α

)

in

B

X

*

converges

to

f

in weak-*

topology

of

X

*

iff

Φ

⁢

(

f

α

)

converges to

Φ

⁢

(

f

)

in product topology, therefore

Φ

is

continuous

and so is its

inverse

Φ

-

1

:

Φ

⁢

(

B

X

*

)

→

B

X

*

.

It remains to show that

Φ

⁢

(

B

X

*

)

is closed. If

(

Φ

⁢

(

f

α

)

)

is a net
in

Φ

⁢

(

B

X

*

)

, converging to a point

d

=

(

d

x

)

x

∈

X

∈

D

, we can define a

function

f

:

X

→

ℂ

by

f

⁢

(

x

)

=

d

x

. As

lim

α

⁡

Φ

⁢

(

f

α

⁢

(

x

)

)

=

d

x

for all

x

∈

X

by

definition

of weak-* convergence, one can easily see that

f

is a

linear functional

in

B

X

*

and that

Φ

⁢

(

f

)

=

d

. This shows that

d

is actually in

Φ

⁢

(

B

X

*

)

and finishes the proof.

Title

proof of Banach-Alaoglu theorem

Canonical

name

ProofOfBanachAlaogluTheorem

Date of creation

2013-03-22 15:10:03

Last modified on

2013-03-22 15:10:03

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

12

Author

Mathprof (13753)

Entry type

Proof

Classification

msc 46B10

Generated on Fri Feb 9 21:36:16 2018 by

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weak* convergence in normed linear space

weak* convergence in normed linear space

•

(

x

n

′

)

⊂

X

′

•

X

a

Banach space

•

∃

x

′

∈

X

′

:

∀

x

∈

X

:

lim

n

→

∞

⁡

x

⁢

(

x

n

′

)

≡

x

n

′

⁢

(

x

)

=

x

′

⁢

(

x

)

.

•

If

X

is

reflexive

(

http://planetmath.org/DualSpace

), then weak*

convergence

is the same as

weak convergence

Note: This is a “seed” entry written using a short-hand format described in

\htmladdnormallink

this FAQhttp://www.ma.utexas.edu/ jcorneli/h/FAQ/.

Title

weak* convergence in

normed linear space

Canonical name

WeakConvergenceInNormedLinearSpace

Date of creation

2013-03-22 14:02:20

Last modified on

2013-03-22 14:02:20

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

9

Author

bwebste (988)

Entry type

Definition

Classification

msc 46B10

Related topic

WeakConvergence

Generated on Fri Feb 9 21:36:18 2018 by

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Banach spaces of infinite dimension do not have a countable Hamel basis

Banach spaces of infinite dimension do not have a countable Hamel basis

A

Banach space

of

infinite

dimension

does not have a

countable

Hamel basis

.

Proof

Let

E

be such space, and suppose it does have a countable Hamel basis, say

B

=

(

v

k

)

k

∈

ℕ

.

Then, by

definition

of Hamel basis and

linear combination

, we have that

x

∈

E

if and only if

x

=

λ

1

⋅

v

1

+

…

+

λ

n

⋅

v

n

for some

n

∈

ℕ

. Consequently,

E

=

⋃

i

=

1

∞

(

span

(

v

j

)

j

=

1

i

)

.

This would mean that

E

is a countable union of proper

subspaces

of finite dimension (they are proper because

E

has infinite dimension), but every finite dimensional proper subspace of a

normed space

is nowhere dense, and then

E

would be

first category

. This is absurd, by the

Baire Category Theorem

.

Note

In fact, the Hamel dimension of an infinite-dimensional Banach space
is always at least the

cardinality

of the

continuum

(even if the

Continuum Hypothesis

fails).
A one-page proof of this has been given by H. Elton Lacey

[

1

]

.

Examples

Consider the set of all real-valued infinite

sequences

(

x

n

)

such that

x

n

=

0

for all but finitely many

n

.

This is a

vector space

, with the known

operations

. Morover, it has infinite dimension: a possible basis is

(

e

k

)

k

∈

ℕ

, where

e

i

⁢

(

n

)

=

{

1

,

if

⁢

n

=

i

0

,

otherwise

.

So, it has infinite dimension and a countable Hamel basis.
Using our result, it follows directly that there is no way to define a norm in this vector space such that it is a

complete metric space

under the induced metric.

References

1

H. Elton Lacey,

The Hamel Dimension of any Infinite Dimensional

Separable

Banach Space is c

,
Amer. Math. Mon. 80 (1973), 298.

Title

Banach spaces of infinite dimension do not have a countable Hamel basis

Canonical

name

BanachSpacesOfInfiniteDimensionDoNotHaveACountableHamelBasis

Date of creation

2013-03-22 14:59:12

Last modified on

2013-03-22 14:59:12

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

21

Author

yark (2760)

Entry type

Result

Classification

msc 46B15

Generated on Fri Feb 9 21:36:19 2018 by

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basic properties of seminorms

basic properties of seminorms

Proposition 1

.

Suppose

p

:

V

→

R

is a

seminorm

on a real (or complex)

vector space

V

.
Then

1.

p

⁢

(

0

)

=

0

,

2.

p

⁢

(

v

)

≥

0

for all

v

∈

V

.

Proof.

Property

1

follows using homogeneity;

p

⁢

(

0

)

=

p

⁢

(

0

⋅

0

)

=

|

0

|

⁢

p

⁢

(

0

)

=

0

.

Property

2

follows using sublinearity and Property 1;

0

=

p

⁢

(

0

)

=

p

⁢

(

v

-

v

)

≤

p

⁢

(

v

)

+

p

⁢

(

-

v

)

=

2

⁢

p

⁢

(

v

)

.

∎

Title

basic properties of seminorms

Canonical name

BasicPropertiesOfSeminorms

Date of creation

2013-03-22 14:38:57

Last modified on

2013-03-22 14:38:57

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Theorem

Classification

msc 46B20

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

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Golab’s theorem

Golab’s theorem

Theorem

. Let

D

be the

unit disc

of a Minkowski plane and let

ℓ

⁢

(

∂

⁡

D

)

denote the Minkowski length (

http://planetmath.org/LengthOfCurveInAMetricSpace

) of the boundary of

D

. Then

6

≤

ℓ

⁢

(

∂

⁡

D

)

≤

8

. The

lower bound

is attained if and only
if

D

is linearly

equivalent

to a

regular hexagon

. The upper bound
is attained if and only if

D

is a

parallelogram

.

Note that 1/2 the

perimeter

of the unit disc is a

constant

between 3 and 4. The special case of the 2-norm yields a constant,
which is known as

π

. So Golab’s theorem is that ”pi” for a
Minkowski plane is always between 3 and 4.

References

GO

S. Golab

, Quelques problèmes métriques de la géometrie de Minkowski,

Trav. l’Acad. Mines Cracovie

6

(1932) 1-79.

PE

C.M. Petty

,

Geometry

of the Minkowski plane, Riv. Mat. Univ. Parma (4)

𝟞

(1955) 269-292.

SC

J.J. Schäefer

, Inner

diameter

, perimeter, and girth of spheres,
Math. Ann.

𝟙𝟟𝟛

(1967) 59-79.

ACT

A.C. Thompson

, Minkowski Geometry, Encyclopedia of Mathematics and its Applications, 63, Cambridge University Press, Cambridge, 1996.

Title

Golab’s theorem

Canonical name

GolabsTheorem

Date of creation

2013-03-22 16:50:33

Last modified on

2013-03-22 16:50:33

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 46B20

Generated on Fri Feb 9 21:36:23 2018 by

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