On the Order Form of the Fundamental Theorems of Asset Pricing ()
1. Some Remarks on Previous Work about the Fundamental Asset Pricing Theorems
The First Fundamental Theorem of Asset Pricing states that the absence of arbitrage for a stochastic process
is equivalent to the existence of an equivalent martingale measure for
. It was shown in [1] that for a locally bounded
-valued semi-martingale
the condition of No Free Lunch with Vanishing Risk is equivalent to the existence of an equivalent local martingale measure for the process
. It was proved in [2] that the local boundedness assumption on
may be dropped under the notion of equivalent
-martingale measure. The work [3] , also discussed in [4] , is still essential in this topic and actually this work’s results rely on what Kreps established as the viable market model consisted by an incomplete market and a linear price system on it. In the present work we are going to resolve the so-called Strictly Positive Extension Property from the financial aspect. The presence of heavy-tails in continuous time models and the possible change of frame from
spaces to Orlicz spaces in order to fit the modelling requirements, oblige us to search for more general versions of the two FTAPs, mostly relied on the geometry of these spaces. Recently, in [5] , a Fundamental Theorem of Asset Pricing and a Super-Replication Theorem in a model-independent framework are both proposed. But these theorems are proved in the setting of finite, discrete time and a market consisting of a risky asset
, as well as options written on this risky asset, too. Notions like the one of the strictly positive projection or that of the filtration are alike the ones met in [6] . A difference between our notion of strictly positive projection and the equivalent notion in [6] is that ours is weaker. That’s because if
implies
, this implies
, because if
, it would be
. An important difference between the article of Troitsky and ours is that we extend the framework of Definitions so as to include cases of non-discrete time spaces. Another one is that we apply these notions in order to provide a new version of the two FTAP, while in [6] an important ordered -space theory of martingales in Banach lattices is developed. Finally, markets subspaces are taken to be sublattices because of the fact that we may include layers of call and put options written on an initial market space, as we remarked in [7] . The present paper is organized as follows: First, we provide some useful notions and definitions and examples for them, as well. Next, we prove the Order Form of the FTAP in the Banach-lattice case and in the next sections we provide the analog of these results in the finite-models case. We also explain the application of our results on the Black-Scholes-Merton model. We also compare them to the Example developed in [4] . The case of non-lattice cones is examined in the last section of the paper, in relation with the classes of reflexive and strongly reflexive cones, mentioned in [8] . The role of the existence of an unconditional basic sequence in a Banach space is also quoted in this section independently from the results provided in [8] , as an important condition for the extraction of results concerning FTAP. This condition is not irrelevant to ( [9] , Th. 1.1), about Lindelöf Properties of weak topology, but here it mainly concerns the construction of a Strictly Positive Projection Operator. On the other side, in the paper [10] ideals of
are used in order to deduce an FTAP-like result ( [10] , Lem. 1), while our results refer to sublattices.
2. Useful Notions and Preliminaries
We consider two periods of time (0 and 1) and a non-empty set of states of the world
which is supposed to be an infinite set. The true state
that the investors face is contained in some
, where
is some σ-algebra of subsets of Ω which gives the information about the states that may occur at time-period 1. A financial position is a
-measurable random variable
. This random variable is the profile of this position at time-period 1. We suppose that the probability of any state of the world to occur is given by a probability measure
. The financial positions are supposed to lie in some subspace
of
, being a Banach lattice.
Definition 1 An incomplete market in
is some sublattice
of
. A complete market in
is some sublattice
of
, such that
.
It is well-known that we define the positive cone
of a subspace
of an ordered vector space to be the set
, where
denotes the positive cone of
.
Definition 2 A positive projection
is a projection, which maps each element of
to some ele- ment of its subspace
, such that
. A positive projection is called strictly positive, if
.
We also recall the notion of random field.
Definition 3 A random field is a map
where
is a Bananch lattice,,
is a topological space and
, for any
. Such a random field
is called associated to the pair
.
We also may provide the notion of the filtration in the frame of random fields:
Definition 4 A filtration associated to the pair
is a net of projections
, where
, where
is a sublattice of E and if
. A is a directed set, by some binary relation
, called direction.
Definition 5 A binary relation
on A is called direction on A, if it is reflexive and transitive on A, while for any
there is a
, such that
.
Definition 6 If
and
, this is denoted by
.
Definition 7 A filtration
is called strictly positive if
.
We also give the definition of the adapted random field under this frame.
Definition 8 A random field
, where
is called adapted to the filtration
, being associated to the pair
if
for any
, where A is a directed subset of
by some binary relation
, which is reflexive, transitive and every pair has an upper bound.
Definition 9 A random field
, where
, has the Martingale Property if it is adapted to a filtration
, being associated to the pair
, while
.
Definition 10 A random field
, where
, has the Strictly Positive Martingale Property
if it is adapted to a filtration
, being associated to the pair
, it has the Martingale Property, while the filtration
is consisted by strictly positive projections.
We give some examples for the previously mentioned notions.
Example 11 If
is a sub-algebra of the
-algebra
of Ω, then since
is a sublattice of
, then
is an incomplete market of financial positions in
.
Example 12 The subspace of partially linear functions M in the space
is a complete market in
, due to the Stone-Weierstrass Theorem. We notice that the partially linear functions defined on
is actually
the sublattice generated by the bi-set of functions
, where
,
. We
notice that in this case, the span of this bi-set is a lattice-subspace of
(see also [11] ).
Example 13 A finite-dimensional sublattice
of
is an incomplete market in
. As a lattice- subspace, it actually has a positive basis with nodes ([11] , Pr. 2.2), hence the equivalent positive projection
is defined as follows:
![]()
where
and
are the nodes of the positive basis of
.
Example 14 A sequence of sublattices
of
characterized by increasing non-terminal parts of the sequence
![]()
which has different terms in the sense
is a filtration of
, since
is the node for the one- dimensional subspace
,
is the set of nodes of the positive basis of
and so on.
Example 15 An increasing net of sub-
-algebras
of Ω, being a non-empty set, where A is a non- empty directed set, induces as it is well-known the existence of a filtration in
, where
is supposed to be a probability space associated to the measurable space
. The relevant net of sublattices is:
![]()
A may denote a set of cardinals, where if we start from a certain cardinal number
, then the cardinality of
-algebra
as a class of objects is at most equal to
and it is surely greater than
.
Example 16 The filtration of the Example 14 is not strictly positive. This holds because if we pick a sublattice
whose positive basis’ nodes is the set:
![]()
If
, this does not imply
if
. For example,
, but
.
Example 17 If
is a Banach lattice with order continuous norm and
is a projection band, namely
, then
is norm-closed. The projection
is strictly positive since it is positive and
, implies that since
, where
and
,
.
and
, hence
and finally x = 0. The same situation is valid for Kantorovich-Banach spaces (or else KB-spaces), in which
. Such examples of spaces are reflexive Banach lattices like
and
-spaces.
Example 18 Let us consider a Banach lattice
which has a Schauder basis:
![]()
which is moreover a positive basis. Also, suppose that:
![]()
are finite-dimensional sublattices of
. Then,
![]()
is a filtration, because
and
, since
is a positive basis of
itself. This is the case for
.
3. Order Versions for the Fundamental Theorems of Asset Pricing
In the proof of the two next Theorems we use the following:
Lemma 19 A positive projection
, where
is a Banach lattice and
is a positive sublattice of it, is a continuous operator.
Proof: Obvious, because every positive operator from a Banach lattice into to a locally solid Riesz space, is continuous.
Theorem 20 (Order 1st Fundamental Theorem of Asset Pricing) Let
be a Banach lattice and
be a sublattice of
. If
admits a strictly positive projection, then every strictly positive and continuous func- tional
, admits a strictly positive, continuous extension on
. Also, if
is a Banach lattice and
is a sublattice of
such that every strictly positive and continuous functional
, admits a strictly positive, continuous extension on
, then
admits a strictly positive projection.
Proof: The adjoint operator of the strictly positive projection
is an injection. Hence
is a continuous, strictly positive functional of
. This is due to the duality:
![]()
For the proof of the opposite, we have the following: We define the projection
as follows.
.
is a positive operator from a Banach lattice into a locally solid Riesz space. Hence it is continuous. By duality for some
strictly positive, continuous functional
of
,
![]()
Hence if we suppose that there is some
such that
, while
. But this leads to a contradiction.
Corollary 21 If
is a Banach lattice which has the Strictly Positive Martingale Property with respect to some filtration
, where
is a directed set. If
such that
, then every strictly positive and continuous functional
, admits a strictly positive, continuous extension on
.
Corollary 22 Let
be a Banach lattice of financial positions and
be an incomplete market, such that
is a market model. If
admits a strictly positive projection, then for every price system
, the market model
is viable.
The existence of a strictly positive projection may be replaced by the Strictly Positive Martingale Property with respect to some filtration in the statement of the above Theorem. The term viable is the one established in the seminal work of D.M. Kreps (see [3] , p. 18-19).
Theorem 23 (Order 2nd Fundamental Theorem of Asset Pricing) Let
be a Banach lattice and
be a dense sublattice of
. If
admits a strictly positive projection, then every strictly positive and continuous functional
, admits a unique strictly positive, continuous extension on
. Also, let
be a Banach lattice and
be a sublattice of E such that M admits a strictly positive projection. Moreover, every strictly positive and continuous functional
, admits a unique strictly positive, continuous extension on
. Then
is dense in
.
Proof: Since M is a dense sublattice of E, the adjoint (linear by the duality
)
operator
of the strictly positive projection
is a surjection. Hence for any
,
there is some
, such that
, or else by duality relations:
![]()
For the converse, we have that for any
, there is some
, such that
, or else by duality relations:
![]()
where
is a strictly positive projection. This implies that
is a surjection, which is equivalent to the fact that
is dense in
.
Corollary 24 If
is a Banach lattice which has the Strictly Positive Martingale Property with respect to some filtration
, where A is a directed set. If
is an element of
such that
is a dense sublattice of
, then every strictly positive and continuous functional
, admits a unique strictly positive, continuous extension on
.
Corollary 25 Let
be a Banach lattice of financial positions and
be a complete market, such that
is a market model. If
admits a strictly positive projection, then for every price system
, the market model
is viable.
The term viable is the one established in the seminal work of D.M. Kreps (see [3] , pp. 18-19).
We may notice that our Theorem does not make any reference to the No -Free Lunch Condition, but it simply extends the No-Arbitrage Property all over the space
. Theorem 23 is the analog of the usual 2nd FTAP, which implies that the (local) Equivalent Martingale Measures’ set of a complete market is a singleton, while under this class of market spaces the uniqueness of the (strictly positive) extension of a price system all over the space of financial positions is achieved under no presence of the No-Free Lunch Condition, too.
Let us see some Examples which confirm the connection of the above Theorems to well-known models of Mathematical Finance.
Example 26 Let
be a probability space endowed with an
-dimensional Brownian motion
, where
. Denote by
the filtration that this Brownian motion
generates, i.e.,
. We assume a financial market consisting of
assets whose prices
are modelled by an
-adapted,
-dimensional Itô process
of the form
where:
![]()
![]()
![]()
![]()
where
is the
-th row
of the
-matrix process
. The process
repre- sents the price of a riskless asset (where
is the interest rate process which is supposed to have bounded values), while the
-th component
, of the process
, represents the evolution of the price of the
-th asset (stock). The price of the riskless asset may be used as numeraire. Suppose that
. If
is a stochastic exponential, then as it is well-known, the following relation holds:
![]()
where
is the probability measure defined on
as follows:
, according to the
Girsanov-Cameron-Martin Theorem. Taking mean values over
we have:
![]()
which in terms of evaluation maps’ values is interpreted as follows:
![]()
The equivalent Riesz pairs are:
![]()
where the strictly positive projection
is
, the strictly positive linear functional
is
and its strictly positive extension
is
. This Example gives also a Hilbert space taste, due to the presence of
-spaces, see also [12] .
Example 27
![]()
holds for the unique possible change of measure
, if the market is complete for example in the Black-Scholes model and this arises indeprendently from the unique solution of the market-price-of-risk equation.
Finally, we may revisit the Example constructed in [4] , in order to quote it.
Example 28 The actual form of the elements of the subspace M of
is described by the following strictly positive projection:
![]()
is a sublattice of
under the usual component-wise ordering. Also,
, while according to
Theorem 3, a strictly positive extension of
all over
exists, through duality relation
.
4. The Finite-State, One Period-Model Case
We will show how the above Theorems 3, 23 are applied in finite -state space models.
Let us consider the two-date market model in which the number of states of the world is denotes by
, while the time-periods are denoted by 0 and 1, respectively. We also consider an incomplete market of primitive assets whose time-period −1 payoffs are the positive, linearly independent vectors
of
, whose span is denoted by
. We suppose that
contains the riskless asset
, while
, which implies
standard incompleteness. We also assume a time-period
, no-arbitrage price
for the pri-
mitive assets. As it is well-known from ([7] , p. 4),
is identified to the sublattice
of
generated by
. We also remind of the following Projection Basis Theorem for sublattices of
, which arises from both ( [13] , Th. 3.7), ( [14] , Th. 9).
Theorem 29 Let
be a
-dimensional subspace of
with
generated by the positive elements
in which the riskless bond 1 is a marketed asset
. Suppose that the range
of the basic function
of the elements
is the finite set
of the simplex
of
(note that
). Suppose that the first
vectors of this set are linearly independent. If we suppose that the vectors
are such that
and
where
where
and
(which are
the vectors indicated by ( [13] , Th. 3.7), then,
1)
.
2) ![]()
3) If
with
and
then the vectors
defined by:
![]()
where
is the
matrix whose columns are the vectors
are a basis of
called projection basis. This basis has the property: The
first coordinates of an element
in the positive basis of
coincide with the coordinates of the expansion of
in the basis
.
Also, according to what is mentioned in [7] about the completion of an incomplete market
by options and by following the notation we introduced,
, where
and
is a maximal set of linearly independent, positive vectors of
. Due to ([7] , Th. 21),
are portfolios of call and put options written on elements of
, especially since
.
The dimension equation which holds in the case of the no-arbitrage price
, is:
![]()
where
denotes the subspace of
generated by the columns of the payoff matrix
of the primitive securities, while
denotes the orthogonal subspace of it. Due to the characterization of the ab- sence of arbitrage in the primitive asset market (see [15] , Th. 9.2), there is at least one
such that
where
. This implies that
in this case, while
if by
we also denote the
matrix whose columns are the vectors
. The last relation arises from
if we suppose that
. Then
and if we denote
, we obtain
the last relation. As it is implied in [7]
is determined by the positive basis
of it.
We also have the following:
Theorem 30 Any
such that
implies a no-arbitrage price
for which the price
of the portfolio
or else the price of the asset
lying in the completion
to be equal to the price of the same asset under
if
, where
are the vectors indicated by the Projection Basis Theorem.
Proof: Consider the vector
. The above vector satisfies the following equalities:
![]()
The definition of the vector
allows us to prove that it is a no-arbitrage price in the subspace generated by the vectors
which is the completion by options
of
. If for a portfolio
the payoff
lies in the positive cone
except
, then:
,
because
. Also, from the Projection Basis Theorem 29, if
, this means that:
.
Hence
in this case, which is equal to the valuation of the portfolio ![]()
of the primitive assets under
. We remind that
is the space of the financial positions, since
is actually equal to this space according to ([7] , Pr. 6).
Theorem 31 (First Order Finite Fundamental Theorem of Asset Pricing) For any subspace
of
, where
and
and
are linearly inde- pendent, every strictly positive linear functional of
has a strictly positive extension on
.
Proof: Every strictly positive functional
defines a no -arbitrage price
on
as follows:
. According to Theorem 30,
for some
such that
is a strictly positive extension of f on
, where
, where
is the support of the vector
of the positive basis of
, see ([7] , Th. 6).
Proposition 32 If we suppose that the vectors of the date-1 payoffs of the primitive assets
are linearly independent and
, then
, where
, except a set of vectors
of Lebesgue measure zero in
.
Proof: In the last part of [7] , a brief proof was given about the fact that resolving markets have the property
. It is also well-known that resolving matrices are in general position, namely the complement of the set of them is a null-set in the vector space of the matrices
, whose entries are real numbers. Hence the super-set of all the
-matrices (markets), such that
where
are linearly independent and they have the property that
are also in general position.
Theorem 33 (Second Order Finite Fundamental Theorem of Asset Pricing) For almost any subspace
of
, where
and
and
are linearly inde- pendent, every strictly positive linear functional of
has a unique strictly positive extension on
.
Proof: Every strictly positive functional
defines a no-arbitrage price
on
as follows:
. According to Theorem 30,
for a unique
.
5. The Finite Multi-Period Model Case
Let us see what happens in the multi-period framework. We consider the event -tree model as it is presented in [15] , according to which there is a finite time -horizon
, a family of partitions F of
such that
and
is thinner than
for any
in the sense
that for any
, there is a
such that
. Then the set
is
the event-tree corresponding to the family of partitions
. Every event-tree
is a model of information re- vealing along the time-periods of
. We also consider
assets (financial contracts) whose payoff vectors are
and if we denote by
the physical number which is equal to the cardinality of the nodes of the event-tree
, these are actually vectors of
. We also suppose that the price vectors of the assets are
, where
if
and the set
denotes the set of nodes of the event-tree corresponding to the time-period
. If we suppose that these price vectors do not provide arbitrage opportunities in the market of the assets
, then since the market is incomplete there is at least one node-price vector
such that
, where
is the payoff matrix of this market as it is indicated in ([15] , Ch. 4). In order to simplify things, we may suppose that
, where
. We also suppose that one of the assets of the market is riskless, or else that for any
which corresponds to the same time-period, its payoff is the same. Also, this asset’s initial price
is equal to 1. The submatrix
for any
is the
-matrix whose rows are the vectors
of
, indicating the payoffs and the ex-payoff price of the J primitive securities at the node
. The cardinality of
is denoted by
.
The market of the securities is complete or as it is usually said the securities’ markets are dynamically complete, if every contingent claim
can be replicated by a portfolio
. In order to understand the next, we remind of the following,
Definition 34 The forward -start call option written on a contingent claim
with exercise- price
at the node
given that
, is equal to:
![]()
Definition 35 The forward -start put option written on a contingent claim
with exercise- price
at the node
given that
, is equal to:
![]()
As a reference for these options we append to ([16] , Par. 9.2).
The market is (dynamically) complete if and only if it is one-period complete for any non-terminal node
, namely if
. Otherwise it is called incomplete. For any
such that
, and moreover there is a non-terminal node
such that for the corresponding sub-
matrix
of
,
holds, we may add forward-start op-
tions of the form
, where
to make it complete. In the
same way we may talk about the completion by options of the span
with re-
spect to the asset
which may be denoted by
for any
.
is the vector of the
Euclidean space
such that
. In a way similar to [7] , the dimension of the
completion
is denoted by
. It is obvious that we may reach a complete market if and only
if
for any
. A question which also arises in this case is how the new assets introduced in
a submarket
with
in order to reach
are priced. The answer is given in the next Theorem, being equivalent to Theorem 29.
Theorem 36 For any submarket
with
and any ![]()
where
, where
is a no-arbitrage price vector for the assets
.
is a price vector which assigns the price
to the portfolio
. Specifi-
cally, the price of the asset
lying in the completion
is equal to the price of the
same asset under
if
, where
are the vectors indicated by the Projection Basis Theorem.
Proof: Consider the vector
, where
. The above vector satisfies the following equalities:
![]()
The definition of the vector
allows us to prove that it is a no-arbitrage price in the subspace generated by the vectors
which is the completion by options
,
.
If for a portfolio
the payoff
lies in the positive cone
except
, then:
,
because
. Also, from the Projection Basis Theorem, if
, this means that
. Hence
in this case, which is equal to the valuation of the
portfolio
of the primitive assets under
. This concludes the proof.
Theorem 37 (First Order Event-Tree Fundamental Theorem of Asset Pricing). For any submarket
with
and any
with
, every strictly
positive linear functional of
has a strictly positive extension on
.
Proof: If
is a strictly positive functional of
, then this implies a no-arbi- trage price
and since
is given,
. The extension of
is
for
some
such that
is a strictly positive extension of f on
, where
, where
is the support of the vector
of the po-
sitive basis of
, see ([7] , Th. 6).
Theorem 38 (Second Order Finite Fundamental Theorem of Asset Pricing) If the market is complete, then for any submarket
, every strictly positive linear functional of
has a unique strictly
positive extension on
.
Proof: Since the market is complete,
and there is a unique
with
, If
is a strictly positive functional of
, then this implies
a no-arbitrage price
and since
is given,
. The unique extension of
is
for some
such that
is a strictly positive extension of ![]()
on
, where
, where
is the support of the vec-
tor
of the positive basis of
, see ([7] , Th. 6), since
is unique.
6. General Cones Revisited
Let us consider a Banach space
of financial positions, partially ordered by a closed cone
, which is not a lattice cone. Such a cone is for example a Bishop-Phepls cone, see ([17] , pp. 126-127), which is well-based and it has also interior points, hence it is not a lattice cone, according to ([17] , Th. 4.4.4). Of course, the set of strictly positive functionals of such a cone has not to be empty. This is the reason due to which the Lindelöf Property mentioned in [9] about the weak topology
defined on a dual system
is important. Of course, there are cones which do not admit continuous strictly positive functionals. Such a cone is the positive cone of an
space, where
is uncountable.
Also, in this section, the definition of (in)completeness are altered.
Definition 39 If M is a infinite-dimensional subspace of E ordered by the cone C, a market is an infinite- dimensional subspace of
, such that
.
Definition 40 A market is incomplete if
, while it is complete if
.
Then, the following versions of the Second and the First Fundamental Theorem of Asset Pricing are deduced, respectively.
Theorem 41 Let
be a Banach space with an unconditional basis. Then a non-lattice one exists, which makes
a complete market and every strictly positive functional of this cone admits a unique strictly positive extension.
Proof: As it is well-known from ([18] , Th. 4.2.22), the cone of the unconditional basis ![]()
makes X a Banach lattice under an equivalent norm. According to ([8] , Th.
5.7) there is a strongly reflexive cone (see [8] , Def. 5.1) C in E+, such that
. Also, since the one-dimensional-subspace projections
are continuous, according to ([18] , Cor. 4.2.26), the operator
is a continuous projection from E ordered by
(which is also the cone of the positive basis)
to
being ordered by
. Also, we notice that
is strictly positive in the sense that
, whenever
. Hence,
may be taken as a strictly positive projection, and consequently we may repeat the proof of Theorem 23.
Theorem 42 Let E be a Banach space with an unconditional basic sequence. Then, for the incomplete market
arising from the basic sequence, there exists a non-lattice cone
, such that
and strictly posi- tive functional of this cone admits a strictly positive extension on
.
Proof: According to ([8] , Cor. 5.8) there is a strongly reflexive cone (see [8] , Def. 5.1)
in
, such that
, while for any uncoditional basic sequence it is well-known that (see [18] , Th. 4.2.22) its cone
makes
a Banach lattice (under an equivalent norm). Also, since the one-dimensional-subspace projec-
tions
are continuous, according to ([18] , Cor. 4.2.26) the operator
is a continuous
projection from
ordered by
(which is also the cone of the positive basis) to
being ordered by
. Also, we notice that P is strictly positive in the sense that
, whenever
. Hence,
may be taken as a strictly positive projection, and consequently we may repeat the proof of Theorem 3.
In the proof of ([8] , Th. 5.7) the strongly reflexive cone’s construction relies exactly on the existence of an unconditional basis for the Banach space E. Then we may understand that the crucial point for the above Theorems is the existence of a basic sequence for the Banach space E. We may remind the seminal work by Bessaga-Pelczynski [19] essentials on this topic.
Appendix
In this Section, we give some essential notions and results from the theory of partially ordered linear spaces which are used in this paper. For these notions and definitions, see ([17] , Ch. 1, Ch. 2, Ch. 3). Let
be a (normed) linear space. A set
satisfying
and
for any
is called wedge. A wedge for which
is called cone. A pair
where
is a linear space and
is a binary relation on
satisfying the following properties:
1)
for any
(reflexive);
2) If
and
then
, where
(transitive);
3) If
then
for any
and
for any
, where
(compatible with the linear structure of
), is called partially ordered linear space. The binary relation
in this case is a partial ordering on
. The set
is called (positive) wedge of the partial ordering
of
. Given a wedge
in
, the binary relation
defined as follows:
![]()
is a partial ordering on
, called partial ordering induced by
on
. If the partial ordering
of the space
is antisymmetric, namely if
and
implies
, where
, then
is a cone.
denotes the linear space of all linear functionals of
, called algebraic dual while
is the norm dual of
, in case where
is a normed linear space.
Suppose that
is a wedge of
. A functional
is called positive functional of
if
for any
.
is a strictly positive functional of
if
for any
. A linear functional
where
is a normed linear space, is called uniformly monotonic functional of
if there is some real number
such that
for any
. In case where a uniformly monotonic func-
tional of C exists, C is a cone.
is the dual wedge of
in
. Also,
by
we denote the subset
of
. It can be easily proved that if C is a closed wedge of a reflexive space, then
. If
is a wedge of
, then the set
is the dual wedge of
in
, where
denotes the natural embedding map from
to the second dual space
of
. Note that if for two wedges
of
,
holds, then
.
If C is a cone, then a set
is called base of C if for any
there exists a unique
such that
. The set
where
is a strictly positive functional of C is the base of
defined by
.
is bounded if and only if
is uniformly monotonic. If
is a bounded base of
such that
then
is called well-based. If
is well-based, then a bounded base of C defined by a
exists. If
then the wedge
is called generating, while if
it is called almost generating. If C is generating, then
is a cone of E* in case where E is a normed linear space. Also,
is a uniformly monotonic functional of C if and only if
, where
denotes the norm-interior of
. If
is partially ordered by C, then any set of the form
where ![]()
is called order-interval of E. If E is partially ordered by
and for some
,
holds, then
is called order-unit of E. If E is a normed linear space, then if every interior point of C is an order-unit of E. If E is moreover a Banach space and
is closed, then every order-unit of E is an interior point of C. The partially ordered vector space E is a vector lattice if for any
, the supremum and the infimum of
with respect to the partial ordering defined by P exist in E. In this case
and
are denoted by
,
respectively. If so,
is the absolute value of
and if E is also a normed space such that
for any
, then E is called normed lattice. If a normed lattice is a Banach space, then it is called Banach lattice. A Banach lattice E whose norm has the property
is called AL-space. A set S in a vector lattice E is called solid if
and
implies
. A solid vector subspace of a vector lattice is called ideal. An ideal
is a sublattice of E, i.e., a subspace of E such that
if
respectively. A net
in a vector lattice E is order convergent to
if
there is a net
in E with
, such that
for each
. This convergence is denoted
by
. A set
in E is order closed if
and
, implies
. If
is also an ideal, then
is called band. A Banach lattice has order continuous norm, if for any net
with
,
holds. A Banach lattice E which is a band in its second dual (in the sense of norm topology) is called Kantorovich-Banach space. If S is a subset of a vector lattice E, then its disjoint complement is the set
. If for a vector lattice E a band B satisfies the property
, then B is called projection band. Finally, if E is a partially ordered Banach space whose positive cone is
, if E has a
Schauder basis
, this basis is called positive basis if and only if
. For
linear lattices and positive bases see in ( [20] , Ch. 8), and [11] , respectively.