First, this work provides an overview of some of the Hahn-Banach type theorems. Of note, some of these extension results for linear operators found recent applications to isotonicity of convex operators on a convex cone. Next, the work investigates applications of the Krein-Milman theorem to representation theory and elements of Choquet theory. A sandwich theorem of intercalating an affine function between and where and are convex, on a finite-simplicial set, is recalled. Its recent topological version is also noted. Here, the novelty is that a finite-simplicial set may be unbounded in any locally convex topology on the domain space. Third, the paper summarizes and comments on recently published applications of a Hahn-Banach extension result for positive linear operators, combined with polynomial approximation on unbounded subsets, to the Markov moment problem. Some applications to concrete spaces are detailed as well. Finally, this work provides a characterization of a finite-dimensional convex bounded subset in terms of the property that any convex function defined on that subset is bounded below. This last property remains valid for a large class of convex operators.
This work generally provides commentary on old and new aspects and applications of Hahn-Banach type results in representations related to the Choquet theory (such as sandwich results on finite-simplicial sets), the Markov moment problem and polynomial approximation on unbounded subsets, and characterization of finite-dimensional convex bounded subsets. All vector spaces appearing in the sequel are real vector spaces. One of the main consequences of the Hahn-Banach theorem is that on a Hausdorff locally convex space , there are many enough linear continuous functionals, which separate the points of On the other hand, recent results require other consequences of a more general Hahn-Banach type theorem. Main such results are stated and discussed in the following sections and accompanied by applications. Versions in the framework of ordered topological vector spaces with normal cones are also emphasized, since concrete spaces have natural such structures. The assumption on order completeness of is motivated by applications of Hahn-Banach type results for extension of linear operators from a subspace of to ; usually, this extension preserves a sandwich condition defined by means of a dominating convex operator and a dominated concave operator. Generally, these operators are defined on arbitrary convex subsets of a real linear space When there is already a given linear order relation on X, the concave operator mentioned earlier is usually defined on and eventually might be null. When this is the case, the result is the positivity of the linear extension, while the convex dominating constraint controls the continuity and determines or evaluates the norm of the linear extension. Such constraints on the extension are motivated by concrete problems mentioned in the Abstract.
In Section 2, this work reviews almost all important versions and applications [1,2,3,4,5,6,7,8,9,10] of the Hahn-Banach theorem, starting with its geometric form  and going to the general forms stated and proved in [8–10]. Such results have found recent applications to the isotonicity of convex operators on convex cones . If are ordered vector spaces, an operator is called isotone if it is monotone increasing:
A similar definition works for isotone operators defined on the entire space (for example, if is a vector lattice, then is sublinear and isotone on ).
In Section 3, this work reviews elements of representations, Choquet’s theory, and related results. This part is covered by references [2,3,12–14]. The first part of Section 3 is dedicated to the important notion of a barycenter of a probability measure on a compact convex subset in a locally convex space Any point of such a subset is the barycenter of a probability measure on K, for which only the behavior on the extreme points of matters. When is metrizable, the representing measure is supported by the extreme points of (Choquet’s theorem). Conversely, any probability measure on an arbitrary compact convex subset has a unique barycenter. This is unique because linear continuous functionals on separate the points of All locally convex spaces involved in this work are assumed to be Hausdorff. To conclude this third part, the recent topological version  of one of the results of  on sandwich theorems over finite-simplicial sets is especially noted. Notably, a convex subset of the real vector space is called finite-simplicial if for any finite-dimensional compact subset there exists a finite-dimensional simplex such that . For example, in any convex cone having a base that is a simplex is an unbounded finite-simplicial set. According to Theorem 3.12, on such a subset, the following result holds (see also , Corollary 3.5 for the proof).
Let be an arbitrary vector space, a finite-simplicial subset, a convex function, a concave function such that on Then, there exists an affine function such that .
Notably, in the aforementioned statement, the dominating function is concave, while the dominated one is convex. This differentiates this particular result from the usual Hahn-Banach type theorems based on the separation of convex subsets. Theorem 3.13 provides a topological version of this result (see , pp. 9 and 10 for the proof).
Section 4 is devoted to another field of applications of Hahn-Banach theorems and other results in Analysis and Functional Analysis, namely, to the classical moment problem. Being given a sequence , of real numbers and a closed subset find a positive regular Borel measure (or a positive Radon measure) on such that
This is an inverse problem, because the measure is not known. Finding means characterizing its existence such that the aforementioned moment conditions are satisfied, studying its uniqueness (called determinacy) and eventually constructing it. All these should be done starting from the known moments This is a full moment problem since it involves all the moments of the measure If we require only
for some fixed natural number we have a truncated moment problem. For the moment problem is called one dimensional, while for , we have a multidimensional moment problem. The references [15–24] concern various aspects of the moment problem. If an upper boundedness condition on is required, we have a Markov moment problem. Such a condition usually controls the norm of the linear positive continuous functional defined by the measure on a function space containing polynomials and compactly supported continuous real-valued function on Usually, such a space is a Banach lattice (for example, an space, The interested reader can find more information on Banach lattices in specialized monographs (see  and a part of ). For the present work, results from [1,11,26] on this topic are sufficient. For more details, see the introductory portion of Section 4. The papers [27,28,29,30] refer mainly to the Markov moment problem. Since the existence of a solution of the moment problem is an extension type result of a linear form defined on polynomials to a larger space, Section 4 is directly related to Hahn-Banach type results of Section 2. In the present work, existence and uniqueness of the solution for the full Markov moment problem are of special interest. The construction of a polynomial solution for the truncated moment problem is proposed in [28,29]. On the other hand, the notion of a moment determinate measure is basic because it leads to the existence and uniqueness of the linear operator solution, also controlling its norm (see [27,29]). An improved version of a result of  on operator-valued Markov moment problem is also stated.
Finally, in Section 5, this work seeks to revisit the result of  on convex operators where is a convex bounded subset and is an order complete vector lattice. In , we proved that any such operator is bounded below on The proof was done by means of the existence of a subgradient of at an arbitrary relative interior point of This way, the strong relationship between convex and linear operators is pointed out once more. Conversely, if is a convex subset of an arbitrary (infinite-dimensional) real vector space such that any real convex function defined on is bounded below, then is finite-dimensional (and bounded). This is presented in Section 5.
The subjects reviewed later and the attached references relate to other fields of analysis and algebra (self-adjoint operators, symmetric matrixes, quadratic forms, fixed point theory, convex analysis, elements of Choquet theory, and polynomial approximation on Cartesian products of unbounded closed intervals).
Section 6 concludes the paper.
2 Various Hahn-Banach type results
The following lemma is the key result for the direct proof of the geometric version of Hahn-Banach theorem.
(See , pp. 45–46). Let be a real topological vector space (t.v.s.) of dimension at least 2. If is an open convex subset and is not an element of then there exists a one-dimensional subspace of not intersecting
Lemma 2.1 and a standard application of Zorn’s lemma yield:
(See , p. 46). Let be a real t.v.s., let be a linear manifold in and let be a nonempty open convex subset of not intersecting Then, there exists a closed hyperplane in containing and not intersecting
Let be a t.v.s., an open convex subset of a vector subspace of such that a continuous linear functional, a convex upper semi-continuous functional such that for all Then, there exists a continuous linear functional , which extends such that for all
To deduce Corollary 2.3 from Theorem 2.2, one applies Theorem 2.2, where stands for , stands for the graph of stands for According to Theorem 2.2, there exists a closed hyperplane in , which contains such that Due to condition cannot be vertical and hence is the graph of a linear functional From the details of this sketch of the proof, it is easy to observe that extends and is continuous (and linear) from to (see also [1, Exercise 6, p. 69]).
The next result holds in locally convex spaces. All such spaces are assumed to be Hausdorff.
(See [1, Theorem 4.2, p. 49]). Let be a t.v.s., whose topology is locally convex. If is a linear form, defined and continuous on a subspace of then has a continuous extension to the entire space
Given and linearly independent elements of a l.c.s. there exist continuous linear forms on such that
The next result is basic in the finite-dimensional convex analysis due to its applications, including the maximum principle for convex functions.
(Carathéodory; see , p. 7). Let be a convex compact subset. Then, any can be written as convex combination of at most extreme points of
A simple proof of Theorem 2.6 (by induction on the dimension ) is given in , pp. 7–8, essentially using Theorem 2.2 stated earlier. Here is a main application of Theorem 2.6 to convex optimization (in particular to linear optimization).
(See [1, Exercise 26, p. 71]). Let be a nonempty compact subset. Then, its convex hull is compact.
(See [3, p. 171). If is a continuous convex real function on a convex compact subset then attains a global maximum at an extreme point of
(The maximum principle , p. 171). Let be a convex subset of If a convex function attains its maximum on at a point from the relative interior of then is constant on
Next, we recall the following basic results, derived from Theorem 2.2.
(First separation theorem , p. 64). Let be a convex subset of a t.v.s. such that and let be a nonempty convex subset of not intersecting the interior of There exists a closed hyperplane separating and if and are both open, separates and strictly.
(Second separation theorem , p. 65). Let be nonempty, disjoint convex subsets of a locally convex Hausdorff space (l.c.s.) such that is closed and is compact. There exists a closed hyperplane in strictly separating and
Let be a l.c.s. and Then, there exists a continuous linear functional such that
The preceding corollary states that the topological dual X* of a l.c.s. X separates the points of X. On the other hand, by the definition of weak topology on a l.c.s. X, any weak closed subset of X is closed in the initial topology on X. For convex closed subsets, the reverse implication holds as well. Namely, we recall the following well-known consequence of Theorem 2.11:
(See , p. 65) Let be a locally convex space and a convex closed subset. Then, is the intersection of all closed half-spaces containing it. In particular, is closed with respect to the weak topology on
The following result (Theorem 2.15) has a natural geometric meaning; it is based on Lemma 2.14 and Theorem 2.10. It is worth noticing that in the latter theorem, if we additionally assume that is open, then is contained in the open half-space defined by (). Before stating Theorem 2.15, we have to review Lemma 2.14, which generalizes the formula for the distance from a point to a hyperplane in that is well known from analytical geometry.
Let be a normed (real) linear space, a closed hyperplane in Then, the distance is given by the following formula:
(See ). Let be a normed linear space, two convex subsets of such that Then, there exists two closed parallel hyperplanes in which separate the subsets and such that
The next key lemma is used in the proof of the main Theorem 2.17 (Krein-Milman).
(See , p. 67). If is a compact, convex subset of a locally convex space, every closed hyperplane supporting contains at least one extreme point of
We recall that, by definition, a closed hyperplane in the locally convex space under attention is supporting if and is contained in one of the two half-spaces defined by A point is called an extreme point of if from the equality implies In other words, cannot be an interior element of any line segment of ends elements of
(Krein-Milman; see , p. 67). Every compact convex subset of a locally convex space is the closed convex hull of its extreme points.
Krein-Milman theorem says that in any compact convex subset of a l.c.s., there are many extreme points, which generate (any element of is the limit of a net whose elements are convex combinations of extreme points of ).
(See [1, Theorem 10.5, p. 68]). If is a compact subset of a locally convex space such that the closed convex hull of is compact, then each extreme point of is an element of
From Theorem 2.6 (Carathéodory), Corollary 2.7, and Theorem 2.18, the following consequence follows:
If is a compact nonempty subset, then its convex hull is compact and Moreover, each point of can be written as convex combination of at most extreme points of
The aforementioned results are more or less deduced from the geometric form of the Hahn-Banach theorem. In most of the cases motivated by further applications, analytic proofs of Hahn-Banach type theorems are more suitable. Here is the first main result, completely proved in [3, pp. 339–340].
(The Hahn-Banach theorem). Let be a vector space, a sublinear functional, a vector subspace a linear functional, such that for all Then, has a linear extension , such that is dominated by on the entire space
(See , p. 340) If is a sublinear functional on a real vector space then for every element , there exists a linear functional such that
(The Hahn-Banach theorem on normed vector spaces; see , p. 341). Let be a vector subspace of the real normed vector space and a continuous linear functional. Then, has a continuous linear extension , with .
(See , p. 341). If is normed vector space, then for each there exists a linear functional on such that and
One of the reasons for using analytic proofs of Hahn-Banach type theorems is that they work not only for extending linear functional but also for operators. As in the case of functional, the proofs of such type results are quite simple, by means of Zorn’s lemma and extension of linear operators from a subspace of the involved domain space , to a space where preserving some constraints on the extension. The codomain of the operators for which Hahn-Banach type theorems hold must be order complete vector spaces, or even order complete vector lattices. We recall that an ordered vector space is a vector space endowed with an order relation, which is compatible with the algebraic structure of a vector space. Namely, the following two properties are satisfied:
We say that such an order relation is linear. If is an ordered vector space, then is a convex cone, called the positive cone of We always assume that the positive cone is generating . An ordered vector space is called order complete (Dedekind complete) if for any upper-bounded subset there exists a least upper bound for in denoted by A vector lattice is an ordered vector space with the property that for any there exists In a vector lattice , for any element , one denotes An ordered Banach space is a Banach space , which is also an ordered vector space, such that the positive cone is closed and the norm is monotone on
A Banach lattice is a Banach space, which is also a vector lattice, such that
Obviously, any Banach lattice is an ordered Banach space. In an ordered Banach space, there exists also the compatibility of the topology defined by the norm with the order relation. There exist ordered Banach spaces that are not lattices. For example, the space of all symmetric matrixes with real coefficients, endowed with the norm
and the order relation , for all is an ordered Banach space, which is not a lattice for Here, the norm is the Euclidean norm of the vector In the same way, if is a real or complex Hilbert space, the real vector space of all self-adjoint operators acting on with the norm and order relation defined similarly to the case of symmetric matrixes, is an ordered Banach space, which is not a lattice (here is replaced by ). Almost all usual function spaces and sequence spaces have natural structures of Banach lattices. On a vector space of real-valued functions defined on a set the usual order relation is: for all For example, if is a compact Hausdorff topological space, the space of all real-valued continuous functions over is a Banach lattice with respect to the aforementioned order relation and usual norm. If we assume that is compact, is connected, nonempty, and not reduced to a singleton, then is not order complete. A particular such a Banach lattice is In other words, the only case when is order complete is that of a totally disconnected space The Lebesgue spaces and the sequence spaces are order complete Banach lattices.
Here is one of the old results on this subject, with many applications to the vector-valued moment problem. Let be an ordered vector space whose positive cone is generating ( Recall that in such an ordered vector space , a vector subspace is called a majorizing subspace if for any there exists such that The following theorem holds. Here is a significant example of a majorizing subspace. Let be a closed unbounded subset and Let be a positive regular Borel measure on with finite moments of all orders. We denote , the vector subspace of all functions for which there exists a polynomial such that on Then, the subspace of all polynomial functions on is a majorizing subspace of The space contains (the subspace of all continuous compactly supported real functions on ), as well as the subspace The subspace is dense in since it contains , which is dense in
(See , Theorem 1.2.1). Let be an ordered vector space whose positive cone is generating, a majorizing vector subspace, an order complete vector space, a positive linear operator. Then, admits a positive linear extension .
We continue with Hahn-Banach type theorems. Now a condition on the operator solution of being dominated by a convex operator defined on a convex subset of the domain space is required. In other words, a generalized Hahn-Banach theorem will be reviewed. The relationship between the next result and its corollary (existence of subgradients of convex operators) will appear clearly. A point of the subset of a vector space is called an (algebraic) interior point of if for each there is a positive such that for The point is said to be an (algebraic) relative interior point of if for each of the affine variety generated by (affine hull of ) there is a positive such that for The set of all interior points of is denoted by A int and the set of all relative interior points by For the next result, see [6, Theorem 2.1, pp. 284–286].
(A generalized Hahn-Banach theorem; see , Theorem 2.1, p. 284). Let be a vector space, a vector subspace, an order complete vector space, a convex subset, a convex operator, a linear operator such that
If then there exists a linear operator such that
(See [6, Corollary 2.7, p. 286]). Let be a vector space, an order complete vector space, a convex subset, a convex operator. If then there exists a linear operator such that
A linear operator satisfying (1) is called a subgradient of at Corollary 2.26 says that a convex operator having as codomain an order complete vector space admits a subgradient at every relative interior point of its domain. This result (with a somewhat different proof) goes back to . The set of all subgradients of at is called the subdifferential of at and is denoted by This is a convex set, and, for convex operators satisfying the hypothesis of Corollary 2.26, is nonempty.
In the results stated earlier, the order relation that naturally exists on concrete spaces does not appear on the domain space in any way. The next theorems take into consideration linear order structures on as well. This way, from now on, we have three conditions on the linear operator solution Namely, must extend a given linear operator defined on a subspace of it is dominated by a given convex operator and dominates a given concave operator . If then the linear extension is positive: Recall that an ordered vector space , which is also a topological vector space, is called an ordered topological vector space if the positive cone is topologically closed. The next result was published by H. Bauer, and independently by I. Namioka, with different proofs, in different journals, in 1957 (for citation of the original sources see [1, p. 227]).
(See [1, Theorem 5.4, p. 227]). Let be an ordered t.v.s. with positive cone and a vector subspace of For a linear form on to have a linear continuous positive extension it is necessary and sufficient that be bounded above on where is a suitable convex 0 − neighborhood in
The next result is motivated by Theorem 2.27 and the discussion preceding it. Subsequently, all theorems are valid for operators. In particular, the corresponding cases of real-valued functionals follow as consequences. In the next theorem, will be a real vector space, an order-complete vector lattice, convex subsets, a concave operator, a convex operator, a vector subspace, and a linear operator. All vector spaces and linear operators are considered over the real field.
(See , Theorem 1). Assume that . The following two statements are equivalent.
There exists a linear extension of the operator such that
There exists convex, and concave operator such that for all
the following implication holds:
It is worth noticing that the extension of Theorem 2.28 satisfies the following conditions: is an extension of is dominated by on , and dominates on Here, the convex subsets are arbitrary, with no restriction on the existence of relative interior points or on their position with respect to the subspace
The following theorems follow more or less directly as corollaries of Theorem 2.28. For details, see [8,9], while for applications to the abstract Markov moment problem, see all the results of . For applications to characterizing the isotonicity of a convex operator over a convex cone, see  (for example, the proof of Theorem 5 of  uses Theorem 2.33 of this article, Theorem 6 of  uses Theorem 2.34 of this article, and Proposition 1 of  applies Theorem 2.30 of this article. The same article  contains a large class of examples of concrete spaces and operators for which the developed theory works. Also, the article  gives a new proof for a known result: any linear positive operator acting between two ordered Banach spaces is continuous. In particular, this theorem works for operators acting between Banach lattices.
(See , Theorem 2). Let be an ordered vector space, an order complete vector space, a vector subspace, a linear operator, and a convex operator. The following two statements are equivalent.
There exists a positive linear extension of such that on
We have for all such that
One observes that in the very particular case when the order relation on is the equality, from Theorem 2.29, one obtains Hahn-Banach extension theorem for linear operators dominated by convex operators. When the convex operator is defined only on the positive cone of one obtains the following variant of Theorem 2.29 (see  and , Theorem 5):
Let be an ordered vector space, an order complete vector space, be a vector subspace, be a linear operator, and be a convex operator. The following two statements are equivalent.
There exists a positive linear extension of such that
We have for all such that
In Theorem 5 of , a direct sharp proof for Theorem 2.30 is pointed out. The next result provides a sufficient condition on the given linear operators for the existence of the linear extensions. When , it has an interesting geometric meaning.
(See ). Let be a locally convex space, an order complete vector lattice with strong order unit and a vector subspace. Let be a convex subset with the following properties:
There exists a neighborhood of the origin such that (that is, by definition, and are distanced);
A is bounded.
Then for any equicontinuous family of linear operators and for any , there exists an equicontinuous family such that
Moreover, if is a convex balanced neighborhood of the origin such that
and if such that and is large enough such that then the following relations hold:
We have denoted by the gauge attached to
The following theorem is also a Hahn-Banach type result (see Theorem 2.29), but is formulated in terms similar to those of the abstract Markov moment problem . However, the condition of the abstract moment problem is replaced by
(Mazur-Orlicz: see , Theorem 5). Let be a preordered vector space, an order complete vector space, , families of elements in respectively in and a sublinear operator. The following two statements are equivalent:
There exists a linear positive operator such that
For any finite subset and any , the following implication holds true
If in addition we assume that is isotone, the assertions (a) and (b) are equivalent to (c), where
for any finite subset and any the following inequality holds:
The next two variants of the same controlled regularity property of some linear operators are also consequence of Theorem 2.28. Recall that a linear operator is called regular if it can be written as a difference of two positive linear operators If is dominated by a given convex operator , we say that we have a controlled regularity for This terminology is motivated by the fact that in the topological framework, is assumed to be continuous and on the entire domain space usually implies the continuity of . Sometimes, the norm of can be evaluated as well.
(See ) Suppose that is an ordered vector space, is an order complete vector lattice, and is a convex operator. Then for any linear operator , the following two statements are equivalent.
There exist two positive linear operators such that
for all in such that
Most of convex operators appearing in applications are defined on the entire domain space. Therefore, we recall the similar statement to that of Theorem 2.33, but for convex operators
(See , Theorem 3). Assume that is an ordered vector space, is an order complete vector lattice and is a convex operator. For any linear operator , the following two statements are equivalent:
There exist two positive linear operators such that
for all in such that
In the end of this section, we state a general constrained extension result, which can be proved as a consequence of Theorem 2.28. Probably, Theorems 2.28 and 2.35 are equivalent.
(See ). Let be a vector space, be an order complete vector lattice, be a vector subspace, be a linear operator, be a convex subset, and be a concave operator. Assume that The following two statements are equivalent.
There exists a linear operator which extends , such that
There exists a convex operator such that for all , the following implication holds:
Moreover, if satisfies the requirements of (b), then the extension of (a) verifies the relation
Since all concrete spaces are endowed with a natural linear order relation, we restate Theorem 2.35 in the framework of ordered vector spaces.
Let be an ordered vector space, be an order complete vector lattice, be a vector subspace, be a linear operator, be a supralinear operator, and be a convex operator. The following two statements are equivalent.
There exists a linear operator , which extends , such that
For all the following implication holds:
Let be as in the statement of Theorem 2.36. Assume that on Then, there exists a linear operator such that
The last result of this section has also been deduced from the general Theorem 2.28. Theorem 2.38 is applied in the proof of Theorem 3.12 of the next section.
(See [10, Theorem 4]). Let be as in Theorem 2.32, two linear operators. Assume also that is a vector lattice. The following two statements are equivalent.
There is a linear operator such that
For any finite subset and any , the following implication holds true:
If is a vector lattice, then assertions (a) and (b) are equivalent to (c), where
for all , and for any finite subset and we have
3 Krein-Milman theorem and elements of representation theory
We start with an interpretation of Carathéodory’s Theorem 2.6 as an integral representation theorem (by means of a discrete measure). Then, by using Krein-Milman Theorem 2.17 and a passing to the limit procedure (eventually involving convergent subnets), one obtains integral representations in terms of arbitrary probability measures. In what follows, is a compact convex nonempty subset of a (Hausdorff) locally convex space For one denotes by the “point mass” at , that is, is the Borel measure, which equals 1 on any Borel subset of , which contains and equals 0 otherwise. According to these comments, if and is contained in an dimensional subspace of there exist extreme points of and in such that Let us denote Then, for any continuous linear form on , one obtains:
Here, we recall that the first equality in (2) is actually the definition of the Dirac measure associated with the point applied to the restriction to of the continuous linear functional on . The conclusion for all linear continuous forms on one reads as represents In the last equality (2), there is an abuse of notation: we denote in two different ways ( and ) the same measure on In what follows, a probability measure on is a nonnegative regular Borel measure on with
Suppose that is a nonempty compact subset of a locally convex space and is a probability measure on A point in is said to be represented by if
for every continuous linear functional on (other terminology: “ is the barycenter of ” and “ is the resultant of ”).
Note that any point is trivially represented by the interesting fact pointed out by (2) is that for a convex compact subset of a finite-dimensional space, each in may be represented by a probability measure, which “is supported” by the extreme points of A similar result holds for arbitrary convex compact metrizable subsets of (see Theorem 3.3).
If is a nonnegative regular Borel measure on the compact Hausdorff space and is a Borel subset of we say that is supported by if
(Choquet). Suppose that is a metrizable compact convex subset of the locally convex space X, and that is an element of Then, there is a probability measure on which represents and is supported by the extreme points of
For the proof of the preceding theorem, see [2, pp. 14–15]. The next result is somehow similar to Choquet’s theorem, without requiring metrizability condition on (see [2, p. 17]).
(Choquet-Bishop-de Leeuw). Suppose that is a compact convex subset of the locally convex space X, and that is in Then, there is a probability measure on , which represents and which vanishes on every Baire subset of , which is disjoint from the set of extreme points of
Theorems 3.3 and 3.4 claim that any point in is the barycenter of a probability measure essentially defined by its behavior on the set of extreme points of The following question arises naturally: does any probability measure on have a barycenter? The answer is affirmative, and, moreover, for a given probability measure on , there exists a unique corresponding barycenter denoted Namely, the following result holds:
(See [3, Lemma 7.2.3, p. 310]). If is a compact convex subset in the locally convex space and is a probability measure on there exists a unique point such that
for all continuous linear functionals on
Since all the locally convex spaces are assumed to be Hausdorff, the uniqueness of follows from the fact that the topological dual X* of separates the points of The next result follows from the more general Theorem 7.2.4 of  and represents the Jensen integral inequality for a barycenter and probability measures.
(Jensen; see ). Suppose that is a probability measure on the convex compact subset of the locally convex space Then,
for all continuous convex functions
Next, we recall some results on the uniqueness of the representing measure. The uniqueness holds if and only if the compact convex subset is a simplex. Before going to infinite-dimensional simplexes, we review the definition of a finite-dimensional simplex. The sets of the form are called polytopes. If are linearly independent, then is called an simplex, with vertices In this case, and any point of has a unique representation as a convex combination of vertices:
The numbers are called the barycentric coordinates of The standard -simplex (or unit -simplex) in is defined by:
We go on with infinite-dimensional simplexes. As is shown in [2, pp. 51–52], for studying a compact convex subset of a locally convex space and see when is a simplex, it is easier to assume that is the base of a convex cone (with vertex at the origin), i.e., and if and only if there exists a unique and in such that Moreover, as discussed in [2, p. 52], whenever a compact convex subset is a base for a cone we can always assume that it is of the form for some closed hyperplane in , which misses the origin.
If a convex set (not necessarily compact) is a base of a cone , we call a simplex if the space is a vector lattice in the ordering induced by .
Let be a compact convex subset; if and are nonnegative regular Borel measures on we write if for all continuous convex functions on where
(See [2, p. 18]). If is a nonnegative measure on then there exists a maximal measure such that
(Choquet-Meyer; see , pp. 56–57). Suppose that is a nonempty compact convex subset of the locally convex space Then, is a simplex if and only if for each point in there is a unique maximal measure on such that for all continuous affine functions
Next, we recall the statement of D.A. Edwards’ separation theorem (Theorem 16.7 of ).
(Edwards). If and are convex upper semicontinuous real-valued functions on a simplex contained in a locally convex space, with then there exists a continuous affine function on such that .
Of note, sandwich-type theorems such as Theorem 3.11 can be proved when the simplex is replaced by a finite-simplicial set, as discussed in . Here, the novelty is that a finite-simplicial set can be unbounded in any locally convex topology on A convex subset of a vector space is called finite simplicial if for any finite-dimensional compact subset there exists a finite-dimensional simplex such that Here are a few examples:
In any convex cone having a base that is a simplex (the corresponding order relation is laticial) is an unbounded finite simplicial set.
In for each the convex cone defined by
has a compact base, but is not finite-simplicial.
Let be an arbitrary infinite or finite-dimensional vector space (of dimension , a non-null linear functional and Then, the sets are finite-simplicial.
Let be as in Example 3), two real numbers such that The set
is not finite-simplicial. From the last two examples, we easily infer that generally the intersection of two finite-simplicial sets is not finite-simplicial.
The following sandwich type result holds true:
(See , Corollary 3.5). Let be an arbitrary vector space, a finite-simplicial subset, a convex function, a concave function such that on Then, there exists an affine function such that .
The proof of Theorem 3.12 is using Theorem 2.38 of Section 2. Next, we state a topological version of Theorem 3.12.
(See , Theorem 4, pp. 8–10). Let be an ordered Banach space. Assume that the positive cone is finite-simplicial and there exists such that contains a balanced and absorbing convex subset. Let be convex continuous functions such that Assume also that Then, there exists a continuous linear functional such that on
4 The moment problem and related results
We recall the classical formulation of the moment problem, under the terms of T. Stieltjes, given in 1894–1895: find the repartition of the positive mass on the nonnegative semi-axis, if the moments of arbitrary orders ( ) are given. Precisely, in the Stieltjes moment problem, a sequence of real numbers is given and one looks for a nondecreasing real function , which verifies the moment conditions:
This is a one-dimensional moment problem, on an unbounded interval. Namely, it is an interpolation problem with the constraint on the positivity of the measure . The numbers are called the moments of the measure . Existence, uniqueness, and construction of the solution are studied. The moment problem is an inverse problem: we are looking for an unknown measure, starting from its given moments. The direct problem might be: being given the measure compute its moments The connection with the positive polynomials and extensions of linear positive functional and operators is quite clear. Namely, if one denotes by the vector space of polynomials with real coefficients, and
where is a finite subset, then the moment conditions are clearly satisfied. It remains to check whether the linear form defined by (3) has nonnegative value at each nonnegative polynomial. If this condition is also accomplished, then one looks for the existence of a linear positive extension of to a larger ordered function space , which contains both and the space of continuous compactly supported functions, then representing by means of a positive regular Borel measure on via Riesz representation theorem or applying Haviland theorem. Usually, the positive linear extension is defined on a Banach lattice of functions. For example, if is a positive regular Borel measure on with finite moments of all orders and one denotes by the vector subspace of defined by contains and all continuous real-valued compactly supported functions on If defined by (3) is a positive (linear) functional on one extends to a linear positive functional on by means of Theorem 2.24 ( is a majorizing subspace in ). Usually, this extension is also continuous on the subspace of In this case, can be extended to a linear continuous functional defined on the entire space via density of in (the subspace of all continuous compactly supported functions on is contained in and is dense in ). If an interval (for example, ℝ, or is replaced by a closed subset of we have a multidimensional moment problem. Passing to an example of the multidimensional real classical moment problem, let us denote
If a sequence is given, one studies the existence, uniqueness, and construction of a linear positive form defined on a function space containing polynomials and continuous compactly supported real functions, such that the moment conditions
are satisfied. Usually, the positive linear form (that is called a solution for the moment problem defined by (4)) can be represented by means of a positive regular Borel measure on In this case, we say that is a representing measure for the sequence , and this sequence is called a moment sequence. Similar definitions and terminology are valid when we replace with an arbitrary closed subset of . When an upper constraint on the solution is required too, we have a Markov moment problem (see the last part of this section). From solutions linear functional, many authors considered linear operators solutions. Of course, in this case, the moments are elements of an ordered vector space (usually is an order complete Banach lattice). The order completeness is necessary to apply Hahn-Banach type results for operators defined on polynomials and having as codomain. The classical moment problem is clearly related to the form of positive polynomials on the involved closed subsets of As it is known, there exist nonnegative polynomials on the entire space which are not sums of squares of polynomials, unlike the case (see , Proposition 13.4, p. 318; see also the comments which precede and follows this result). The analytic form of positive polynomials on closed intervals is crucial in solving classical moment problems. Such results are useful in characterizing the existence of a positive solution by means of signatures of quadratic forms. In the case of the Markov moment problem, approximation of nonnegative compactly supported continuous functions (with their support contained in a closed unbounded subset ) by special nonnegative polynomials on that subset, having known analytic form, is very important. For the multidimensional Markov moment problem on Cartesian products of closed unbounded intervals, this method works, provided that each interval is endowed with a moment determinate positive regular Borel measure. Recall that a measure is called determinate (moment determinate, or simply determinate) if it is uniquely determined by its classical moments, or, equivalently, by its values on polynomials. A moment sequence is called determinate if it has only one representing measure. If a sequence has a representing measure supported on a compact subset then is determinate thanks to the Weierstrass approximation theorem. We start reviewing existence of a solution for the simplest classical one-dimensional moment problems: the Hamburger moment problem (when ), Stieltjes moment problem (when ), and Hausdorff moment problem (when ). In the sequel, the following notations are used: is the vector space of all real-valued compactly supported continuous functions defined on is the convex cone of all functions in , which take nonnegative values at each point of is the convex cone of all polynomial functions with real coefficients, which are nonnegative on
(Hamburger’s theorem: see , Theorem 3.8, p. 63). For a real sequence , the following statements are pairwise equivalent.
The sequence is a Hamburger moment sequence, that is, there is a nonnegative Radon measure on such that and
The sequence is positive semidefinite, i.e., for all and , we have
All Hankel matrices are positive semidefinite.
defined by (3) is a positive linear functional on that is, for
(See , p. 65). For a real sequence , the following statements are pairwise equivalent.
is a Stieltjes moment sequence, that is, there is a nonnegative Radon measure on such that and
For all and , we have
All Hankel matrixes are positive semidefinite.
Theorem 4.1 (respectively 4.2) gives necessary and sufficient conditions for a sequence of real numbers to be an -moment sequence (respectively an -moment sequence). Next, we go on with the corresponding problem on (the Hausdorff moment problem).
(See , p. 66). For a real sequence the following statements are pairwise equivalent:
is a moment sequence.
Next, we go on with the problem of determinacy. A Hamburger moment sequence is determinate if it has a unique representing measure, while a Stieltjes moment sequence is called determinate if it has only one representing measure supported on [0,∞). The Carleman theorem contains a powerful sufficient condition for determinacy.
(See , Theorem 4.3, pp. 80–81). Suppose that is a positive semidefinite sequence. The following assertions hold.
If satisfies the Carleman condition
then is a determinate Hamburger moment sequence.
If in addition is positive definite and
then is a determinate Stieltjes moment sequence.
The following theorem of Krein consists in a sufficient condition for indeterminacy (for measures given by densities).
(Krein condition: see , Theorem 4.14, pp. 85–86). Let be a nonnegative Borel function on Suppose that the measure defined by is a Radon measure on R and has finite moments for all
then the moment sequence is indeterminate.
Next, we give new checkable sufficient conditions on distributions of random variables that imply Carleman condition, ensuring determinacy. Consider two random variables with values in with values in . Assume that both and belong to the class and let and be the corresponding densities. All moments of are assumed to be finite. The symbol used later has the usual meaning of “monotone increasing.”
(See , Theorem 1, p. 498: Hamburger case). Assume that the distribution of is symmetric on and continuous and strictly positive outside an interval such that the following conditions hold:
Under these conditions, satisfies Carleman’s condition, and hence, it is determinate.
(See , Theorem 2, p. 498: Stieltjes case). Assume that the density of is continuous and strictly positive on for some such that the following conditions hold:
Under these conditions, satisfies Carleman’s condition, and hence, it is determinate.
The distribution function having as density satisfies the conditions of Theorem 4.7; hence, it is determinate.
Going back to the existence problem for a solution, we consider the multidimensional case, which is much more complicated than the one-dimensional moment problem. The main reason is that the analytic form of nonnegative polynomials on closed subsets of is generally not known in terms of sums of squares of polynomials. A case when this difficulty can be solved is that of semi-algebraic compact subsets of . Here is one of the main results on this subject. If is a sequence of real numbers, one denotes by the linear functional defined on by
where is a finite subset and are arbitrary real coefficients. Let be a finite subset of where is the real vector space of all polynomials with real coefficients, of real variables Then, the closed subset given by
is called a semi-algebraic set. The following result was proved for compact semi-algebraic sets (see  Theorem 1.4, and  Theorem II.2.4 for related or more general results). On the other hand, important results on resolution of the moment problem on any compact (not necessarily semi-algebraic) subset with nonempty interior in had been proved in  (see , Theorems 1, 2, and 4). The expression of positive polynomials on such a compact is also deduced in Theorem 4 of .
(See ). Let be a compact semi-algebraic set as defined earlier. Then, there is a positive Borel measure supported on such that
if and only if
(See ). With the aforementioned notations, if is such that for all in the semi-algebraic compact defined by (5), then is a finite sum of special polynomials of the form
for some and
The next results of this section are based on polynomial approximation on unbounded subsets, also using Hahn-Banach type results. We start by recalling the following key approximation lemma.
(See , Lemma 3). Let be an unbounded closed subset, and let be an M-determinate measure on (with finite moments of all natural orders). Then, for any