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Order unit normed linear spaces are a special type of regularly ordered normed linear spaces and therefore the first section is a short collection of the fundamental results on this type of normed linear spaces. The connection between order unit normed linear spaces and base normed linear spaces within the category of regularly ordered normed linear spaces is described in Section 2, and Section 3 at last, contains the results on Banach limits in an arbitrary order unit normed linear space. It is shown that the original results on Banach limits are valid for a greater range.

Most, if not all, publications where Banach limits are investigated take place in an order unit normed real linear space. Order unit normed linear spaces are a special type of regularly ordered normed linear spaces and therefore the first section is a short collection of the fundamental results on this type of normed linear spaces, for the reader's convenience. The connection between order unit normed linear spaces and base normed linear spaces within the category of regularly ordered normed linear spaces is described in Section 2, and Section 3 at last, contains the results on Banach limits in an arbitrary order unit normed linear space. It is shown that the original results on Banach limits are valid in a for greater range. For a further generalisation of vector valued Banach limits in a different direction we refer to a recent paper of R.Armario, F. Kh. Garsiya-Pacheko and F. Kh Peres-Fernandes [

An ordered normed linear space

(Ri 1) For

(Ri 2) For x Î E with

hold. (see [

Lemma 1. Let, for an ordered linear space

(Ri 3) For

(Ri 4) For any

holds.

Proof. The proof is straightforward. Condition (Ri 2) implies that

hence (Ri 3) is proved for

(Ri 3) implies that

Because of (Ri 3), for

In [

Definition 1. If

The set

is a linear space by the obvious operations. One introduces the cone

which is obviously proper and generates

Lemma 2. Let

holds.

Proof. For

and

hence

□

Now, we proceed to define the norm

Proposition 1. For regularly ordered normed spaces

Proof. The proof that

Using

and, multiplying by −1

Adding (i) and (ii) yields

hence

and

Now

hence

□

In the following

The order unit normed linear spaces are a special type of regularly ordered normed linear spaces , as are the base normed linear spaces [

Definition 2. For two order unit normed linear spaces

and

Proposition 2. Let

i)

ii)

Proof. (1) Let

Let

Then

follows, i.e.

which proves that

Now

(ii) This follows from (i) (see [

Corollary 1. For order unit normed linear spaces

is a base-normed ordered linear space with base

Proof. That

□

Remark 1. If

Definition 3. The order unit normed linear spaces together with the linear mappings

There is an equally important subcategory of Reg-Ord, the category of based normed linear spaces.

Definition 4. A base normed ordered linear space “base normed linear space” for short, is a regular ordered linear space

The elements of

is a base normed space of special mappings from

What remains in this connection is to investigate special morphisms particularly adapted to these subcategories between spaces belonging to two different of these subcategories Ord-Unit and BN-Ord. We start this with investigating the intersection of these subcategories.

Proposition 3. Let

Proof. If

Hence, the isomorphism is

It should be noted that this isomorphism is an isomorphism in the category Ord- Unit of order unit normed spaces and also in BN-Ord. So, loosely speaking,

Now the “general connection” between Ord-Unit and BN-Ord is investigated via the morphisms:

Proposition 4. If

Proof. Define

This is a slightly different version of the proof of Theorem 1 in Ellis [

Surprisingly a corresponding result also holds if

Proposition 5. If

Proof. Define

where

that is

For

Obviously

from which

□

It is interesting that by defining the subspaces

There are different ways to generalize the structure of

For the introduction of Banach Limits we first prove, following a proof method of W. Roth in [

Theorem 6. (Hahn-Banach Theorem for Order Unit Spaces) Let

i)

ii)

iii) For any

iv)

v) For any

Then there exists a positive linear functional

a)

b)

c)

d)

for

Proof. Define

Obviously

Let

As

holds.

If

Let

As obviously

Define for

As, for

Taking

implying

Now, the remaining equations in the assertion will be proved for

contributing

to

contributing

to the definition of

To show the invariance of

contributing

to

An inequality

and

as contribution to

Verbatim, this proof carries over to the equation

A new function is now introduced by

If

For

and

follows which implies in, particular,

Taking

in particular

follows i.e. monotonicity.

Consider now, for

then

i.e.

Now for

and

The mapping

is, for fixed

holds because for

We now show that

contributing

to

Hence

This implies

The proof of the remaining two equations of the assertions follows almost verbatim this pattern of proof and one gets:

and (6) implies

because of (6) and the minimality of

Now, looking again at the definition (4) of

which together with (1) yields

Now, for

and

and since

which implies

Banach limits are almost always defined as continuous extensions of a continuous linear functional in an order unit normed space. Hence, for the introduction of Banach limits we need Theorem 6 in a continuous form. Surprisingly Theorem 6 already contains all the necessary continuity conditions as the following Corollary shows:

Corollary 2. Let the assertions (i)-(v) of Theorem 6 be satisfied and put

i)

ii)

iii) Any

Proof. i): Obviously,

ii):

implying the continuity of

It is remarkable that with respect to the continuity properties, the continuity of

Definition 5. With the notations of Corollary 2 any such

One defines

Proposition 7. For

i)

ii)

iii)

Proof. i): Let

all abstract convex combinations, then, for

ously

satisfied, too.

ii): One first proves that

iii): Obviously,

□

Because

is a proper cone and

is a base normed ordered linear space. To simplify notation, we will write

Theorem 8. If the norm induced by

is a compact, base normed Saks space (see [

Proof. As

The result of Theorem 8 is essentially the definition of a functor from any category with objects satisfying the assertions of Theorem 6 to the category of compact, base normed Saks spaces ( [

The main result of the paper offers a Hahn-Banach theorem for order unit normed spaces (Theorem 6) from which novel conclusions on Banach limits are drawn. The result of Theorem 8 gives rise to the definition of a functor which goes from any category with objects satisfying the assertions of Theorem 6 into the category of compact, base normed Saks spaces.

Pallaschke, D. and Pumplün, D. (2016) Banach Limits Revisited. Advances in Pure Mathematics, 6, 1022-1036. http://dx.doi.org/10.4236/apm.2016.613075