000 04131nam a22004935i 4500
001 978-3-0348-0478-3
003 DE-He213
005 20140220082836.0
007 cr nn 008mamaa
008 121029s2013 sz | s |||| 0|eng d
020 _a9783034804783
_9978-3-0348-0478-3
024 7 _a10.1007/978-3-0348-0478-3
_2doi
050 4 _aQA319-329.9
072 7 _aPBKF
_2bicssc
072 7 _aMAT037000
_2bisacsh
082 0 4 _a515.7
_223
100 1 _aCobzaş, Ştefan.
_eauthor.
245 1 0 _aFunctional Analysis in Asymmetric Normed Spaces
_h[electronic resource] /
_cby Ştefan Cobzaş.
264 1 _aBasel :
_bSpringer Basel :
_bImprint: Birkhäuser,
_c2013.
300 _aX, 219 p. 1 illus. in color.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aFrontiers in Mathematics,
_x1660-8046
505 0 _aIntroduction.- 1. Quasi-metric and Quasi-uniform Spaces. 1.1. Topological properties of quasi-metric and quasi-uniform spaces -- 1.2. Completeness and compactness in quasi-metric and quasi-uniform spaces.- 2. Asymmetric Functional Analysis -- 2.1. Continuous linear operators between asymmetric normed spaces -- 2.2. Hahn-Banach type theorems and the separation of convex sets -- 2.3. The fundamental principles -- 2.4. Weak topologies -- 2.5. Applications to best approximation -- 2.6. Spaces of semi-Lipschitz functions -- Bibliography -- Index.
520 _aAn asymmetric norm is a positive definite sublinear functional p on a real vector space X. The topology generated by the asymmetric norm p is translation invariant so that the addition is continuous, but the asymmetry of the norm implies that the multiplication by scalars is continuous only when restricted to non-negative entries in the first argument. The asymmetric dual of X, meaning the set of all real-valued upper semi-continuous linear functionals on X, is merely a convex cone in the vector space of all linear functionals on X. In spite of these differences, many results from classical functional analysis have their counterparts in the asymmetric case, by taking care of the interplay between the asymmetric norm p and its conjugate. Among the positive results one can mention: Hahn–Banach type theorems and separation results for convex sets, Krein–Milman type theorems, analogs of the fundamental principles – open mapping, closed graph and uniform boundedness theorems – an analog of the Schauder’s theorem on the compactness of the conjugate mapping. Applications are given to best approximation problems and, as relevant examples, one considers normed lattices equipped with asymmetric norms and spaces of semi-Lipschitz functions on quasi-metric spaces. Since the basic topological tools come from quasi-metric spaces and quasi-uniform spaces, the first chapter of the book contains a detailed presentation of some basic results from the theory of these spaces. The focus is on results which are most used in functional analysis – completeness, compactness and Baire category – which drastically differ from those in metric or uniform spaces.  The book is fairly self-contained, the prerequisites being the acquaintance with the basic results in topology and functional analysis, so it may be used for an introduction to the subject. Since new results, in the focus of current research, are also included, researchers in the area can use it as a reference text.
650 0 _aMathematics.
650 0 _aFunctional analysis.
650 0 _aOperator theory.
650 0 _aTopology.
650 1 4 _aMathematics.
650 2 4 _aFunctional Analysis.
650 2 4 _aApproximations and Expansions.
650 2 4 _aOperator Theory.
650 2 4 _aTopology.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9783034804776
830 0 _aFrontiers in Mathematics,
_x1660-8046
856 4 0 _uhttp://dx.doi.org/10.1007/978-3-0348-0478-3
912 _aZDB-2-SMA
999 _c96292
_d96292