construction of Banach limit using limit along an ultrafilter

construction of Banach limit using limit along an ultrafilter

Construction of Banach limit using limit along an ultrafilter

The

existence

of

Banach limit

is proved in

mathematical analysis

usually by

Hahn-Banach theorem

. (This proof can be found e.g. in

[

5

]

,

[

2

]

or

[

4

]

.) Here we
will show another approach using limit along a filter. In fact we
define it as an

ℱ

-limit of

(

y

n

)

, where

(

y

n

)

is the
Cesàro mean of the

sequence

(

x

n

)

and

ℱ

is an arbitrary

ultrafilter

on

ℕ

.

Theorem 1

.

Let

F

be a free ultrafilter on

N

. Let

(

x

n

)

be a

bounded

(

http://planetmath.org/Bounded

) real
sequence. Then the

functional

φ

:

ℓ

∞

→

R

φ

⁢

(

x

n

)

=

ℱ

⁢

-

⁢

lim

⁡

x

1

+

…

+

x

n

n

is a Banach limit.

Proof.

We first observe that

φ

is defined. Let us denote

y

n

:=

x

1

+

…

+

x

n

n

. Since

(

x

n

)

is bounded, the
sequence

(

y

n

)

is bounded as well. Every bounded sequence has a
limit along any ultrafilter. This means, that

φ

⁢

(

x

n

)

=

ℱ

⁢

-

⁢

lim

⁡

y

n

exists.

To prove that

φ

is a Banach limit, we should verify its

continuity

, positivity, linearity, shift-invariance and to verify
that it extends limits.

We first show the shift-invariance. By

S

⁢

x

we denote the sequence

x

n

+

1

and we want to show

φ

⁢

(

S

⁢

x

)

=

φ

⁢

(

x

)

. We observe that

x

1

+

…

+

x

n

n

-

(

S

⁢

x

)

1

+

…

+

(

S

⁢

x

)

n

n

=

x

1

+

…

+

x

n

n

-

x

2

+

…

+

x

n

+

1

n

=

x

1

-

x

n

+

1

n

. As the sequence

(

x

n

)

is bounded, the last
expression

converges

to 0. Thus

φ

⁢

(

x

)

-

φ

⁢

(

S

⁢

x

)

=

ℱ

⁢

-

⁢

lim

⁡

x

1

-

x

n

+

1

n

=

0

and

φ

⁢

(

x

)

=

φ

⁢

(

S

⁢

x

)

.

The rest of the proof is relatively easy, we only need to use the
basic properties of a limit along a filter and of Cesàro mean.

Continuity:

∥

x

∥

≤

1

⇒

|

x

n

|

≤

1

⇒

|

y

n

|

≤

1

⇒

|

φ

⁢

(

x

)

|

≤

1

.

Positivity and linearity follow from positivity and linearity of

ℱ

-limit.

Extends limit: If

(

x

n

)

is a

convergent sequence

, then its
Cesàro mean

(

y

n

)

is

convergent

to the same limit.
∎

References

1

B. Balcar and P. Štěpánek,

Teorie množin

,
Academia,
Praha, 1986 (Czech).

2

C. Costara and D. Popa,

Exercises in

functional analysis

,
Kluwer,
Dordrecht, 2003.

3

K. Hrbacek and T. Jech,

Introduction

to

set theory

,
Marcel Dekker,
New York, 1999.

4

T. J. Morisson,

Functional analysis: An introduction to

Banach space

theory

, Wiley, 2000.

5

Ch. Swartz,

An introduction to functional analysis

, Marcel
Dekker, New
York, 1992.

Title

construction of Banach limit using limit along an ultrafilter

Canonical name

ConstructionOfBanachLimitUsingLimitAlongAnUltrafilter

Date of creation

2013-03-22 15:32:29

Last modified on

2013-03-22 15:32:29

Owner

kompik (10588)

Last modified by

kompik (10588)

Numerical id

8

Author

kompik (10588)

Entry type

Application

Classification

msc 03E99

Classification

msc 40A05

Related topic

BanachLimit

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