tailieunhanh - 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. | TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 19, SOÁ T6- 2016 An approximate Hahn-Banach-Lagrange theorem Nguyen Dinh International University, VNU – HCM Tran Hong Mo Tien Giang University (Received on November 5th 2015, accepted on November 21th 2016) ABSTRACT 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 (., the sublinear INTRODUCTION AND PRELIMINARY 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. 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, ., 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 .

• Vật lý
• An approximate Hahn Banach Lagrange theorem
• The Hahn Banach Lagrange
• Hahn Banach Lagrange theorem
• The asymptotic Farkas type
• The celebrated Hahn Banach theorem