Advances in Pure Mathematics
Vol.06 No.13(2016), Article ID:72955,15 pages
10.4236/apm.2016.613075
Banach Limits Revisited*
Diethard Pallaschke1, Dieter Pumplün2
1Institute of Operations Research, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
2Faculty of Mathematics and Computer Science, Fern Universität Hagen, Hagen, Germany
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: November 13, 2016; Accepted: December 20, 2016; Published: December 23, 2016
ABSTRACT
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.
Keywords:
Order Unit Normed Spaces, Base Normed Spaces, Banach Limits
1. 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] .
2. Regularly Ordered Normed Linear Spaces
An ordered normed linear space with order
norm
and order cone
is called regularly ordered iff the cone
is
-closed and proper and
is a Riesz norm, i.e. if
(Ri 1) For implies
i.e.
is absolutely mo- notone, and
(Ri 2) For x Î E with there exists a
with
and
hold. (see [2] [3] ).
Lemma 1. Let, for an ordered linear space with proper and
-closed cone
(Ri 1) hold. Then each of the following two conditions is equivalent to (Ri 2)
(Ri 3) For and
there exists a
such that
and
hold.
(Ri 4) For any
holds.
Proof. The proof is straightforward. Condition (Ri 2) implies that generates
If (Ri 2) holds, then for
and
there is
with
hence (Ri 3) is proved for
(Ri 3) implies that generates
and (Ri 1) implies
Because of (Ri 3), for and
there is a
such that
and
for any
and hence
proving (Ri 4). 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
Definition 1. If are regularly ordered linear spaces a bounded linear mapping
is called positive iff
holds. A bounded linear mapping is called regular iff it can be expressed as the difference of two positive linear mappings [3] .
The set
is a linear space by the obvious operations. One introduces the cone
which is obviously proper and generates One often writes
as abbreviation for
and consequently calls an
positive and writes
if
for
in
i.e.
The positive part of the unit ball in a regularly ordered space
with norm
is denoted by
Lemma 2. Let be regularly ordered normed linear spaces with norm
and cone
If
and
denotes the usual supremum norm, then
holds.
Proof. For with
there is
with
which implies
and
hence
□
Now, we proceed to define the norm in the space
by
(*)
Proposition 1. For regularly ordered normed spaces is a Riesz norm on
and makes
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
and
Let
and
with
Then
follows and
(i)
Using and
one obtains in the same way
and, multiplying by −1
(ii)
Adding (i) and (ii) yields
hence
and
Now yields
i.e.
is a norm. If
then
hence and
is a Riesz norm because of Lemma 1, (Ri 4) and the definition of
.
is obviously
-closed and therefore also
-closed because 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
has as an adjoint, the tensor product [3] .
3. 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 with order unit
define
and
Proposition 2. Let be order unit spaces with order units
Then
i) is a
-closed convex base of
and
the
-closed unit ball of
in the supremum norm
ii) is a
-closed proper subcone of
Proof. (1) Let and
with
Then, for
follows which implies
and
i.e.
showing that
is
-closed. Now,
and
imply
i.e.
and even
because
Hence
follows, even
Let be a convex combination of
Then
follows, i.e.
which proves that is convex.
Now with
and
implies
and
i.e.
is a
-closed base of
and
(ii) This follows from (i) (see [7] , 3.9 p. 128). □
Corollary 1. For order unit normed linear spaces
is a base-normed ordered linear space with base and base norm denoted by
and
are closed in the base norm
Proof. That 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
-closed (see Proposition 2) and because the
-topology is weaker than the
-topology (see Proposition 2 and [7] , 3.8.3, p. 121).
□
Remark 1. If is a Banach space, with the norm
because
are Banach spaces, then
is superconvex (see [3] [6] ) and
is a base normed Banach space (see [3] [4] [7] ).
Definition 3. The order unit normed linear spaces together with the linear mappings with
constitute the category Ord-Unit of order- unit 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 with proper closed cone
and norm
which is induced by a base
of
(see [4] [7] ). If
are base normed linear spaces, put
The elements of are monotone mappings,
is a base set in
and it is
-closed. Let
denote the proper closed cone generated by
.
is a base normed space of special mappings from to
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.
Proposition 3. Let be a regular ordered normed linear space. Then
is a base and order unit norm iff
is isomorphic to
by a regular positive isomorphism.
Proof. If is the order unit and if we omit the index
at the norm, then trivially
and
hold. Let
and assume
As
(see [7] ),
holds and
follows or
which implies
because
is additive on
This implies
and hence
which gives a contradiction. Therefore
and the assertion follows as
and
Hence, the isomorphism is defined by
□
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 is a base normed and
an order unit normed linear space, then
is an order unit normed linear space.
Proof. Define by
and extend
positive linearly by
for
to
which can be uniquely extended to
a monotone, linear mapping in
in the usual way. Obviously,
with
the Reg-Ord norm, as
is a positive mapping. Take a
with
i.e.
and hence
for
or
whence
for
For arbi- trary
and
follows implying
for
or
This means, for arbitrary
that
This shows that
is an order unit in
Denoting the order unit norm by
follows. □
This is a slightly different version of the proof of Theorem 1 in Ellis [7] .
Surprisingly a corresponding result also holds if Ord-Uni and
BN-Ord
Proposition 5. If is an order unit and
is a base normed ordered linear space, then Reg-Ord
. is a base normed ordered linear space.
Proof. Define
where denotes the order unit of
One shows first that
is a base set. For this, let
i.e.
and
implying for
that is and
hence
this implies that
For and
obviously
and
is convex. Besides, the above proof shows, that any
can be written as
with
Obviously and if
with
and
implying
from which follows because of
and finally
□
It is interesting that by defining the subspaces and
of
for order unit or base normed spaces
respectively, one gets a number of results which for the bigger space
have either not yet been proved or were more difficult to prove because the assumptions for
are weaker (see [10] [11] ). The Pro- positions 4 and 5 are an exception because here the general space
has the special structure of an order unit or base normed spaces, respectively.
There are different ways to generalize the structure of in many fields of mathematics. In analysis one is primarly interested in aspects of order, norm and convergence. Now, essentially,
with 1, the usual order and the absolute value (considered as a norm) forms the intersection
which both generalize
in different (dual) directions. The above results seem to indicate that the order unit spaces are at least as important as generalizations of
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
and
4. Banach Limits
For the introduction of Banach Limits we first prove, following a proof method of W. Roth in [12] , Theorem 2.1, a special variant of the Hahn-Banach Theorem for order unit normed linear spaces.
Theorem 6. (Hahn-Banach Theorem for Order Unit Spaces) Let be an order unit normed space with order unit
ordering cone
and norm
and let the following conditions be satisfied:
i) is a sublinear monotone function with
ii) is a surjective positive linear mapping.
iii) For any the set mapping
is a right inverse of
with
is monotone and
iv) is a muliplicative group
of positive automorphisms of
v) For any and for every
,
and
hold.
Then there exists a positive linear functional with:
a) and
b)
c)
d)
for and
Proof. Define
Obviously holds, hence
A partial order “
” is defined in
by putting, for
Let be a non-empty, with respect to “
” totally ordered subset and define
As for all
is well defined and finite and
holds.
If then
for all
and hence for all
follows, i.e.
is monotone.
Let then for all
and hence
As obviously for
it follows that
is sublinear. Also
trivially satisfies the conditions a)-d) as well as
Consequently,
and
is a lower bound of
in
Zorn’s Lemma now implies the existence of (at least) one minimal element in
with respect to
which will be denoted by
Define for
As, for
(1)
Taking and
in the defining equation of
yields
(2)
implying
(2a)
Now, the remaining equations in the assertion will be proved for Take the inequality
from the defining equation of
then
contributing
to Conversely,
leads to
contributing
to the definition of and one gets
(3a)
To show the invariance of under
start with
from
Then
contributing
to
An inequality of
leads to
and
as contribution to. Hence
(3b)
Verbatim, this proof carries over to the equation
(3c)
A new function is now introduced by
(4)
If then
is
and therefore
For,
implies
and yields
because of the definition of
. Hence
and
(5)
follows which implies in, particular, for
as
is positive.
Taking and
one has
(6)
in particular If
and
in (4) then
(7)
follows i.e. monotonicity.
Consider now, for
then
i.e.
Now for
and
The mapping
is, for fixed bijective, therefore
(8)
holds because for one has
So
is sublinear.
We now show that also satisfies the equations of the assertions of the theorem. Take
from the defining set of
Then
contributing
to Conversely,
contributing
to
yields by applying
Hence
This implies
(9)
The proof of the remaining two equations of the assertions follows almost verbatim this pattern of proof and one gets:
and (6) implies Hence
is proved which implies
(10)
because of (6) and the minimality of
Now, looking again at the definition (4) of and putting
and
one gets
(11)
which together with (1) yields and in combination with (1) and (10) gives
(12)
Now, for
and
and since and
are superlinear, one has
which implies that is
is superadditive and because of (12) positively homogeneous, i.e. superlinear. This implies that
is linear because 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 and
for
Then
i) and
is an isometrical order unit normed subspace of
which is closed.
ii) is continuous and
is a positive, continuous linear functional with
iii) Any satisfying Theorem 6 is a positive, continuous linear extension of
Proof. i): Obviously, and, hence,
For
holds and this is an inequality in
and
which proves i).
ii): and
are both monotone,
sublinear and
superlinear. As for
holds,
is linear and positive on
. If
then
and
s. th. norm of
is
and
is in 0 continuous hence also for any
and
Hence, we get for any
in Theorem 6
implying the continuity of and
even
because
In particular, this holds also for
□
It is remarkable that with respect to the continuity properties, the continuity of and
do not play any role.
Definition 5. With the notations of Corollary 2 any such is called a Banach limit of
One defines
of course, depends on the parameters
but in order to make the notation for the following not too cumbersome we will mostly omit them and write simply
Proposition 7. For the following statements hold:
i) is a convex subset of
of the base normed Banach space
the dual space of
ii) is weakly-*-closed, weakly closed and also
-closed, where
denotes the usual dual norm of
of
iii) is a superconvex base set contained in
.
Proof. i): Let be the set of
all abstract convex combinations, then, for
obvi-
ously as
for
and obviously all other equations in Theorem 6 are
satisfied, too.
ii): One first proves that is
-closed, because from this , the other two assertions of ii) then follow at once. If
and there is a
with
for
then
i.e.
follows for
which implies
for
and
for
Analogously, one shows the other equations in Theorem 6 for
because they hold for the
iii): Obviously, holds, hence
is bounded and because of ii)
-closed. Then, it is a general result that
is superconvex (see [13] , Theorem 2.5).
□
Because is as a subset of
trivially a base set
is a proper cone and
is a base normed ordered linear space. To simplify notation, we will write instead of
in the following.
Theorem 8. If the norm induced by in
is denoted by
and the topology induced by the weak-*-topology
in
by
then
is a compact, base normed Saks space (see [13] , Theorem 3.1) and an isometrical subspace of
Proof. As is a weakly-*-closed base set and a subset of
which is weakly-*-compact because of Alaoglu-Bourbaki it is also weakly-*-compact and the space
generated by the closed cone
(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.
5. 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 functor which goes from any category with objects satisfying the assertions of Theorem 6 into the category of compact, base normed Saks spaces.
Cite this paper
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
References
- 1. Armario, R., Garcia-Pacheco, F.J. and Pérez-Fernández, F.J. (2013) On Vector-Valued Banach Limits. Functional Analysis and Its Applications, 47, 82-86. (In Russian) (Transl. (2013) Functional Analysis and Its Applications, 47, 315-318.)
- 2. Davis, R.B. (1968) The Structure and Ideal Theory of the Predual of a Banach Lattices. Transactions of the AMS, 131, 544-555.
https://doi.org/10.1090/S0002-9947-1968-0222604-8 - 3. Min, K.Ch. (1983) An Exponential Law for Regular Ordered Banach Spaces. Cahiers de Topologie et Géometrie Differentielle Catégoriques, 24, 279-298.
- 4. Wong, Y.C. and Ng, K.F. (1973) Partially Ordered Topological Vector Spaces. Oxford Mathematical Monographs, Clarendon Press, Oxford.
- 5. Pumplün, D. (1999) Elemente der Kategorientheorie. Spektrum Akademischer Verlag, Heidelberg, Berlin.
- 6. Pumplün, D. (2002) The Metric Completion of Convex Sets and Modules. Results in Mathematics, 41, 346-360.
https://doi.org/10.1007/BF03322777 - 7. Jameson, C. (1971) Ordered Linear Spaces. Lecture Notes in Mathematics (Volume 141), Springer, Berlin.
- 8. Pumplün, D. (1995) Banach Spaces and Superconvex Modules. In: Behara, M., et al. (Eds.), Symposia Gaussiana, de Gruyter, Berlin, 323-338.
https://doi.org/10.1515/9783110886726.323 - 9. Pumplün, D. and Röhrl, H. (1989) The Eilenberg-Moore Algebras of Base Normed Spaces. In: Banaschewski, Gilmour, Herrlich, Eds., Proceedings of the Symposium on Category Theory and its Applications to Topology, University of Cape Town, Cape Town, 187-200.
- 10. Ellis, E.J. (1966) Linear Operators in Partially Ordered Normed Vector Spaces. Journal of the London Mathematical Society, 41, 323-332.
https://doi.org/10.1112/jlms/s1-41.1.323 - 11. Klee Jr., V.L. (1954) Invariant Extensions of Linear Functionals. Pacific Journal of Mathematics, 4, 37-46.
- 12. Roth, W. (2000) Hahn-Banach Type Theorems for Locally Convex Cones. Journal of the Australian Mathematical Society Series A, 68, 104-125.
https://doi.org/10.1017/S1446788700001609 - 13. Pumplün, D. (2011) A Universal Compactification of Topological Positively Convex Sets and Modules. Journal of Convex Analysis, 8, 255-267.
- 14. Pumplün, D. (2003) Positively Convex Modules and Ordered Normed Linear Spaces. Journal of Convex Analysis, 41, 109-127.
NOTES
*Dedicated to Reinhard Börger, a brilliant and enthusiastic mathematician full of new ideas.