Banach Limits Revisited

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.


Introduction
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 [1].

Regularly Ordered Normed Linear Spaces
An ordered normed linear space E with order ' , ′ ′′ ≤ norm ( ) C E is called regularly ordered iff the cone ( )  Moreover, (Ri 4) obviously implies (Ri 2) which completes the proof.□ In [3] K. Ch.Min introduced regularly ordered normed spaces as a natural and canonical generalization of Riesz spaces.A crucial point in this generalization was the definition of the corresponding homomorphisms compatible and most closely related to the structure of these spaces, such that, in addition, the set of these special homomorphisms is again a regularly ordered normed linear space in a canonical way.This is done by are regularly ordered linear spaces a bounded linear mapping A bounded linear mapping is called regular iff it can be expressed as the difference of two positive linear mappings [3].
The set Reg Ord , : : regular linear mapping is a linear space by the obvious operations.One introduces the cone which is obviously proper and generates ( ) Reg Ord , .

E E −
One often writes 1 0 x ≥ as abbreviation for ( ) x C E ∈ and consequently calls an ( ) ( ) .
Lemma 2. Let i E be regularly ordered normed linear spaces with norm i □ and cone ( ), 1, 2.   i and g ∞ denotes the usual supremum norm, then Proof.For 1 1 x E ∈ with □ Now, we proceed to define the norm * □ in the space ( ) Proposition 1.For regularly ordered normed spaces 1 2 , , E E * □ is a Riesz norm on ( ) Reg Ord , E E − and makes ( ) Reg Ord , E E − a regularly ordered normed linear space.For holds and in general Proof.The proof that * □ is a seminorm is straightforward.In order to show that * ∞ ≤ □ □ one starts with ( ) Using 0 g f − ≥ and 1 1 0 y x − ≥ one obtains in the same way ( ) ( ) ( ) and, multiplying by −1 ( ) ( ) ( ) Adding (i) and (ii) yields ( ) ( ) ( ) □ is a Riesz norm because of Lemma 1, (Ri 4) and the definition of ( ) * .
( ) ( ) In the following * □ will always denote this norm of regular linear mappings.
Note that Reg-Ord is a symmetric, complete and cocomplete monoidal closed category and the inner hom-functor

Order Unit and Base Ordered Normed Linear Spaces
The order unit normed linear spaces are a special type of regularly ordered normed linear spaces , as are the base normed linear spaces [3] [4].For investigating a special type of mathematical objects, however, it is always best to use the type of mappings most closely related to the special structure of the objects (the Bourbaki Principle).
Hence, for investigating order unit normed spaces we do not look at the full subcategory of Reg-Ord generated by the order unit normed spaces but introduce a more special type of regular linear mappings.The same method, by the way, has been successful for another type of regularly ordered spaces, namely the base normed (Banach) spaces (cp.[3] [5] [6]).
Definition 2. For two order unit normed linear spaces i E with order unit , 1, 2, , : Reg Ord , , 0 and , E E be order unit spaces with order units 1 2 , .

Bs E E ∉
(ii) This follows from (i) (see [7], 3.9 p. 128).□ Corollary 1.For order unit normed linear spaces , 1, 2, is a base-normed ordered linear space with base ( ) , C E E and ( ) Ord Unit , E E − is a base normed space follows from Proposition 2 and the definition.That base and cone are base normed closed follows from the fact that they are 0 .□ -closed (see Proposition 2) and because the 0 .□ -topology is weaker than the 0 .□ -topology (see Proposition 2 and [7], 3.8.3,p. 121).
( ) ). Definition 3. The order unit normed linear spaces together with the linear mappings Ord Unit , f E E ∈ − constitute the category Ord-Unit of orderunit normed linear spaces which is a not full subcategory of Reg-Ord.
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 E with proper closed cone ( ) The elements of ( ) , Bs E E are monotone mappings, ( ) , Bs E E is a base set in ( ) , C E E denote the proper closed cone generated by , is a base normed space of special mappings from 1 E to 2 .E The base normed linear spaces and these linear mappings form the not full subcategory BN-Ord of Reg-Ord (see [6] [8] [9]), which is therefore a closed category.
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.
, , be a regular ordered normed linear space.Then E □ is a base and order unit norm iff ( ) ( ) is the order unit and if we omit the index E at the norm, then ( ) ( ) 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 1 E is a base normed and 2 E an order unit normed linear space, then ( ) : Bs E e ε = and extend ε positive linearly by ( ) : which can be uniquely extended to  b Bs E ∈ or ( ) ( ) ( ) ( ) ( ) ( ) This shows that ε is an order unit in ( ) Reg Ord , .

E E −
Denoting the order unit norm by □ This is a slightly different version of the proof of Theorem 1 in Ellis [7].Surprisingly a corresponding result also holds if 1 E ∈ Ord-Uni and 2 E BN-Ord Proposition 5.If 1 E is an order unit and 2 E is a base normed ordered linear space, then Reg-Ord ( ) , E E . is a base normed ordered linear space.Proof.Define Reg Ord , and where 1 e denotes the order unit of 1 E One shows first that ( ) , Bs E E is a base set.
For this, let ( ) ( ) ( ) . f e Bs E ∈ this implies that ( ) , Bs E E is convex.Besides, the above proof shows, that any ( ) ( ) ) ( ) Ord Unit , E E − and ( ) Reg Ord , E E − for order unit or base normed spaces , , E E respectively, one gets a number of results which for the bigger space ( ) Reg Ord , E E − have either not yet been proved or were more difficult to prove because the assumptions for ( ) Reg Ord , E E − are weaker (see [10] [11]).The Propositions 4 and 5 are an exception because here the general space ( ) Reg Ord , E E − has the special structure of an order unit or base normed spaces, respectively.
There are different ways to generalize the structure of R in many fields of mathematics.In analysis one is primarly interested in aspects of order, norm and convergence.Now, essentially, R with 1, the usual order and the absolute value (considered as a norm) forms the intersection

Banach Limits
For the introduction of Banach Limits we first prove, following a proof method of W.  x x   Define for 0 :

T y z T y T z x y z E
Taking 1 0 x x = and 2 0 x = in the defining equation of 0 α yields ( ) ( ) ( ) Now, the remaining equations in the assertion will be proved for Take the inequality ( ) x S x x ≤ + from the defining equation of ( ) ( ) to the definition of ( ) ( ) To show the invariance of ( ) Verbatim, this proof carries over to the equation A new function is now introduced by If and yields ( ) follows which implies in, particular, ( ) in particular ( ) ( ) ( ).
Consider now, for , 1, 2, holds because for 0, σ = one has ( ) ( ) So µ  is sublinear.We now show that µ  also satisfies the equations of the assertions of the theorem.Take from the defining set of ( ).
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 .12) and satisfies all the equations in the assertion, which completes the proof.□ 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 implying the continuity of µ and 1, µ ≤ even In particular, this holds also for 0 .λ □ It is remarkable that with respect to the continuity properties, the continuity of , , , be the set of all abstract convex combinations, then, for ( ) Ban Lim , , ,  (see [7], Theorem 3.8.3,[13], Theorem 3.2, [14]) is a compact, base normed Saks space.The last assertion is obvious.□ 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 ( [13], Theorem 3.1).This functor will be investigated by the authors in a forthcoming paper.

Summary
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 which goes from any category with objects satisfying the assertions of Theorem 6 into the category of compact, base normed Saks spaces.
Ri 1) hold.Then each of the following two conditions is equivalent to (Ri 2proof is straightforward.Condition (Ri 2) implies that ( )C E generates E If (Ri 2) holds, then for x E ∈ = + because □ is additive on ( ).C E This implies 0 d = and hence e b = which gives a contradiction.Therefore □ the Reg-Ord norm, as ε is a positive mapping.Take a ≤ i.e.
generalize R in different (dual) directions.The above results seem to indicate that the order unit spaces are at least as important as generalizations of R as the base normed spaces while in many publications the latter type seems to play the dominant role.Propositions 4 and 5 are particularly interesting because the hom-spaces have a special structure if the arguments do not belong to the same of the two subcategories Reg Ord − and BN Ord − x y ≤ then ( ) ( ) s x s y ≤ for all s O ∈ and hence for all ( ) ( ) 0 0 , x y s x s y ≤ ≤ follows, i.e. 0 s is monotone.Let , x y E ∈ then for all s O ∈ ( ) ( ) ( ) ( ) 0 s x y s x y s x s y + is a lower bound of O in .p MZorn's Lemma now implies the existence of (at least) one element in p M with respect to ≤ which will be denoted by .µ because of(12) positively homogeneous, i.e. superlinear.This implies that ( ) x µ is linear because of (

5 .
With the notations of Corollary 2 any such µ is called a Banach limit of 0 .λ is a compact, base normed Saks space (see[13], Theorem 3.1) and an isometrical subspace of -*-closed base set and a subset of ( )Bs E′which is weakly-*-compact because of Alaoglu-Bourbaki it is also weakly-*-compact and the space and proper and E□ is absolutely mo-E □ -closed cone The positive part of the unit ball in a regularly ordered space E with norm is a base normed ordered linear space.To simplify notation, we will write ( )