Banach limit

Banach limit

Consider the set

c

0

of all

convergent

complex-valued

sequences

{

x

⁢

(

n

)

}

n

∈

ℕ

.
The limit

operation

x

↦

lim

n

→

∞

⁡

x

⁢

(

n

)

is a

linear functional

on

c

0

, by the usual limit laws.
A

Banach limit

is, loosely speaking, any linear functional that generalizes

lim

to apply to non-convergent sequences as well. The formal definition follows:

Let

ℓ

∞

be the set of

bounded

complex-valued sequences

{

x

⁢

(

n

)

}

n

∈

ℕ

, equipped
with the sup norm.
Then

c

0

⊂

ℓ

∞

, and

lim

:

c

0

→

ℂ

is a linear functional.
A

Banach limit

is any

continuous

linear functional

ϕ

∈

(

ℓ

∞

)

*

satisfying:

i

ϕ

⁢

(

x

)

=

lim

n

⁡

x

⁢

(

n

)

if

x

∈

c

0

(That is,

ϕ

extends

lim

.)

ii

∥

ϕ

∥

=

1

.

iii

ϕ

⁢

(

S

⁢

x

)

=

ϕ

⁢

(

x

)

, where

S

:

ℓ

∞

→

ℓ

∞

is the shift operator defined by

S

⁢

x

⁢

(

n

)

=

x

⁢

(

n

+

1

)

. (Shift invariance)

iv

If

x

⁢

(

n

)

≥

0

for all

n

, then

ϕ

⁢

(

x

)

≥

0

. (Positivity)

There is not necessarily a unique Banach limit. Indeed, Banach limits are often constructed
by extending

lim

with the

Hahn-Banach theorem

(which in turn invokes the

Axiom of Choice

).

Like the limit superior and limit inferior, the Banach limit can be applied for situations
where one wants to algebraically manipulate limit equations or

inequalities

,
even when it is not assured beforehand
that the limits in question exist (in the classical sense).

1

Some consequences of the definition

The positivity condition ensures that the Banach limit of a real-valued sequence is real-valued,
and that limits can be compared:
if

x

≤

y

, then

ϕ

⁢

(

x

)

≤

ϕ

⁢

(

y

)

.
In particular, given a real-valued sequence

x

, by comparison with the sequences

y

⁢

(

n

)

=

inf

k

≥

n

⁡

x

⁢

(

k

)

and

z

⁢

(

n

)

=

sup

k

≥

n

⁡

x

⁢

(

k

)

,
it follows that

lim inf

n

⁡

x

⁢

(

n

)

≤

ϕ

⁢

(

x

)

≤

lim sup

n

⁡

x

⁢

(

n

)

.

The shift invariance allows any finite number of terms of the sequence to be neglected when taking the Banach limit, as is possible with the classical limit.

On the other hand,

ϕ

can never be multiplicative, meaning that

ϕ

⁢

(

x

⁢

y

)

=

ϕ

⁢

(

x

)

⁢

ϕ

⁢

(

y

)

fails.
For a counter-example, set

x

=

(

0

,

1

,

0

,

1

,

…

)

; then we would have

ϕ

⁢

(

0

)

=

ϕ

⁢

(

x

⋅

S

⁢

x

)

=

ϕ

⁢

(

x

)

⁢

ϕ

⁢

(

S

⁢

x

)

=

ϕ

⁢

(

x

)

2

, so

ϕ

⁢

(

x

)

=

0

,
but

1

=

ϕ

⁢

(

1

)

=

ϕ

⁢

(

x

+

S

⁢

x

)

=

ϕ

⁢

(

x

)

+

ϕ

⁢

(

S

⁢

x

)

=

2

⁢

ϕ

⁢

(

x

)

=

0

.

That

ϕ

is continuous means it is

compatible

with limits in

ℓ

∞

.
For example, suppose that

{

x

k

}

k

∈

ℕ

⊂

ℓ

∞

,
and that

∑

k

=

0

∞

x

k

is absolutely convergent in

ℓ

∞

.
(In other words,

∑

k

=

0

∞

∥

x

k

∥

∞

<

∞

.)
Then

ϕ

⁢

(

∑

k

=

0

∞

x

k

)

=

∑

k

=

0

∞

ϕ

⁢

(

x

k

)

by

continuity

.
Observe that this is just the

dominated convergence theorem

, specialized
to the case of the

counting measure

on

ℕ

, in disguise.

2

Other definitions

In some definitions of the Banach limit, condition (i) is replaced
by the seemingly weaker condition that

ϕ

⁢

(

1

)

=

1

—
the Banach limit of a

constant

sequence is that constant.
In fact, the latter condition together with shift invarance implies condition (i).

If we restrict to real-valued sequences,
condition (ii) is clearly redundant, in view of the other conditions.

Title

Banach limit

Canonical name

BanachLimit

Date of creation

2013-03-22 15:23:00

Last modified on

2013-03-22 15:23:00

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

7

Author

stevecheng (10074)

Entry type

Definition

Classification

msc 46E30

Classification

msc 40A05

Related topic

AlmostConvergent

Related topic

ConstructionOfBanachLimitUsingLimitAlongAnUltrafilter

Related topic

ConstructionOfBanachLimitUsingLimitAlongAnUltrafilter2

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