An approximate Hahn-Banach-Lagrange theorem

In this paper, we proved a new extended version of the Hahn-Banach-Lagrange theorem that is valid in the absence of a qualification condition and is called an approximate HahnBanach-Lagrange theorem. This result, in special cases, gives rise to approximate sandwich and approximate Hahn-Banach theorems. These results extend the Hahn-Banach-Lagrange theorem, the sandwich theorem in [18], and the celebrated Hahn-Banach theorem. The mentioned results extend the original ones into two features: Firstly, they extend the original versions to the case with extended sublinear functions (i.e., the sublinear functions that possibly possess extended real values). Secondly, they are topological versions which held without any qualification condition. Next, we showed that our approximate HahnBanach-Lagrange theorem was actually equivalent to the asymptotic Farkas-type results that were established recently [10]. This result, together with the results [5, 16], give us a general picture on the equivalence of the Farkas lemma and the HahnBanach theorem, from the original version to their corresponding extensions and in either nonasymptotic or asymptotic forms. INTRODUCTION AND PRELIMINARY It is well-known that the Farkas lemma for convex systems is equivalent to the celebrated Hahn-Banach theorem [16]. In the last decades, many generalized versions of the Farkas lemma have been developed (see [3, 5, 4, 9, 11, 15, 17], and, in particular, the recent survey [7]). For the generalizations of non-asymptotic Farkas lemma, i.e., the versions of Farkas-type results were hold under some qualification condition. It was shown in [5] that these versions are equivalent to some extended versions of the Hahn-Banach theorem. A natural question arises: Are there any similar results for generalized asymptoic/sequential Farkas lemmas and certain types of extended HahnBanach theorems? This paper is aimed to answer this question. Fortunately, the answer is affirmative, and so the result in this paper can be considered as a counter part of [5] concerning versions of sequential Farkas lemmas and the socalled approximate Hahn-Banach-Lagrange theorems (which are extended versions of the Hahn-Banach theorem). In this paper, we establish a new extended version of Hahn-Banach-Lagrange theorem which extends the original one in [5, 18], and it is valid in the absence of a regularity condition. It is called the approximate Hahn-Banach-Lagrange theorem. The results then gives rise to an approximate sandwich theorem and an approximate HahnBanach theorem in the manner as in [5]. The generalization of these reults in comparison with [5, 18] is twofold: firstly, they extend the original version to the case with extended sublinear functions (i.e., the sublinear functions which possibly possess extended real values); secondly, Science & Technology Development, Vol 19, No.T6-2016 Trang 170 in contrast to [5], they are topological versions which hold without any qualification condition. The paper can be considered as a continuation of the previous ones (of the authors and their coauthors) [5, 10, 12]. Some tools and some ideas of generalizations to Hahn-Banach-Lagrange theorem and to real-extended sublinear functions are modifications of the one in [5] to adapt to the case where no qualification condition is assumed. Let X and Y be locally convex Hausdorff topological vector spaces (lcHtvs in short), with their topological dual spaces X  and , Y respectively. The only topology we consider on * , X Y  is the w -topology. For a set , A X   the closure of A w.r.t. the weak -topology is denoted by cl . A The indicator function of A is denoted by A i , i.e.,   0 A i x  if , x A    A i x   if \ . x X A  Let , B C be two subsets of some locally convex Hausdorff topological vector space. We say that B is closed regarding C if (cl ) B C B C    (see [1], [5]). Let : { }. f X    ¡ The effective domain of f is the set     dom : : < . f x X f x    The function f is proper if dom . f  The set of all proper, lower semi-continuous (lsc in short) and convex functions on X will be denoted by  . X  The epigraph of f is     epi : ( , ) : . f x X f x       ¡ The Legendre-Fenchel conjugate of f is the function : : { } f X       ¡ ¡ defined       , , . sup x X f x x x f x x X     

functions that possibly possess extended real values).Secondly, they are topological versions which held without any qualification condition.Next, we showed that our approximate Hahn-Banach-Lagrange theorem was actually equivalent to the asymptotic Farkas-type results that were established recently [10].This result, together with the results [5,16], give us a general picture on the equivalence of the Farkas lemma and the Hahn-Banach theorem, from the original version to their corresponding extensions and in either nonasymptotic or asymptotic forms.

INTRODUCTION AND PRELIMINARY
It is well-known that the Farkas lemma for convex systems is equivalent to the celebrated Hahn-Banach theorem [16].In the last decades, many generalized versions of the Farkas lemma have been developed (see [3,5,4,9,11,15,17], and, in particular, the recent survey [7]).For the generalizations of non-asymptotic Farkas lemma, i.e., the versions of Farkas-type results were hold under some qualification condition.It was shown in [5] that these versions are equivalent to some extended versions of the Hahn-Banach theorem.A natural question arises: Are there any similar results for generalized asymptoic/sequential Farkas lemmas and certain types of extended Hahn-Banach theorems?This paper is aimed to answer this question.Fortunately, the answer is affirmative, and so the result in this paper can be considered as a counter part of [5] concerning versions of sequential Farkas lemmas and the socalled approximate Hahn-Banach-Lagrange theorems (which are extended versions of the Hahn-Banach theorem).
In this paper, we establish a new extended version of Hahn-Banach-Lagrange theorem which extends the original one in [5,18], and it is valid in the absence of a regularity condition.It is called the approximate Hahn-Banach-Lagrange theorem.The results then gives rise to an approximate sandwich theorem and an approximate Hahn-Banach theorem in the manner as in [5].The generalization of these reults in comparison with [5,18] is twofold: firstly, they extend the original version to the case with extended sublinear functions (i.e., the sublinear functions which possibly possess extended real values); secondly, Trang 170 in contrast to [5], they are topological versions which hold without any qualification condition.The paper can be considered as a continuation of the previous ones (of the authors and their coauthors) [5,10,12].Some tools and some ideas of generalizations to Hahn-Banach-Lagrange theorem and to real-extended sublinear functions are modifications of the one in [5] to adapt to the case where no qualification condition is assumed.
Let X and Y be locally convex Hausdorff topological vector spaces (lcHtvs in short), with their topological dual spaces BC be two subsets of some locally convex Hausdorff topological vector space.We say that proper, lower semi-continuous (lsc in short) and convex functions on X will be denoted by  .We assume by convention: where Sy respectively, then " S -convex" means "affine", "convex", "concave" or "arbitrary", respectively.
It can be easily verified that if (see [5]).
The organization of the paper is as follows: In the next section, Section 2, we recall two new versions of sequential Farkas lemma for coneconvex systems and sublinear-convex systems established in [10].In Section 3, we establish the so-called approximate Hahn-Banach-Lagrange theorem, a topological and asymptotic extended version of the original algebraic one in [18,19,20].Versions of approximate sandwich theorem and approximate Hahn-Banach theorem are derived from this approximate Hahn-Banach-Lagrange theorem.The last section, Section 4, we show that our new approximate Hahn-Banach-Lagrange theorem is actually equivalent to the asymptotic Farkas-type results that were established recently in [10].This equivalence can be considered as the last piece of the whole picture on the equivalence of the Farkas lemma and the Hahn-Banach theorem for which the other pieces are the equivalence of non-asymptotic extended convex Farkas lemmas and extended Hahn-Banach-Lagrange theorem established in [5], and the one between the linear Farkas lemma and the celebrated Hahn-Banach theorem [16].

Sequential Farkas lemma for convex systems
In this section we will recall the sequential Farkas lemmas for convex systems in [10] which hold without any qualification condition: the asymptotic version of the Farkas lemma for systems which is convex w.r.t. a convex cone and the one for systems which is convex w.r.
(iv)there exists a net () From the previous theorem, it is easy to see that under some closedness conditions, one gets back stable Farkas lemma established recently in [5] (see [10]).

Approximate Hahn-Banach-Lagrange theorem
In this section we establish the so-called approximate Hahn-Banach-Lagrange theorem, a topological and asymptotic extended version of original algebraic version in [18], [19], and [20].An approximate sandwich theorem and an approximate Hahn-Banach theorem are derived from this approximate Hahn-Banach-Lagrange theorem.
It is worth mentioning that these extended versions of Hahn-Banach-Lagrange theorem, sandwich theorem, and Hahn-Banach theorem extended the original ones in two features: they extend the original version to the case with extended sublinear functions and, in contrast to [5], they are topological versions which hold without any qualification condition.
We will maintain the notations used in Section 2.

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Proof.Let :   ¡¡ be the function defined by ( ) =    for all .  ¡ It is clear that  is proper convex continuous function and The conclusion follows from Theorem 2.2.Firstly, (2.2) follows from the assumption (3.1).
[ ( , , , , )  ).Moreover, by (3.5), (3.3) can be rewritten as ( ) ( )( ) , Combing this inequality and (3.6), we get 1 2 3 ) ( )( ) , Taking liminf in the last inequalities and taking the fact that ( The converse implication is trivial.The proof is complete. As a consequence of Theorem  It is well-known that the original (linear) Farkas lemma for convex systems is equivalent to the celebrated Hahn-Banach theorem [16].For the generalizations of non-asymptotic Farkas lemma, i.e., the versions of Farkas-type results that hold under some qualification conditions, it was shown in [5] that these versions are equivalent to some extended versions of the Hahn-Banach theorem.In this section, we establish the counter part of [5] concerning versions of sequential Farkas lemmas and the so-called approximate Hahn-Banach-Lagrange theorem just obtained in Section 3. Concretely, we show that two versions of sequential Farkas lemma for cone-convex systems and for sublinear-convex systems in [10] On the other hand, if (i) in Theorem 2.1 holds, i.e., , ( )  ,, 2) By the definition of limit superior, for any > 0,  there exists 0 iI  such that 1 2 3 0 ( ) ( ) ( ) ( ) , .
Indeed, similar to the proof of Corollary 3.1, the set in (2.1) is closed in the product space XX  ¡ and (3.1) in Theorem 3.1 also holds as *,