zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The Goldstine theorem for asymmetric normed linear spaces. (English) Zbl 1185.54028
If X is a linear space, then a function q:X + is called an asymmetric norm on X if for all x,yX and r + , x=0 if and only if q(x)=q(-x)=0, q(rx)=rq(x) and q(x+y)=q(x)+q(y). It follows that the function q s :X defined by d s (x)=max(q(x),q(-x)) is a norm on X. The authors study the dual and bidual spaces of (X,q) and of (X,q s ) and establish a characterization of reflexive asymmetric normed linear spaces.
MSC:
54E35Metric spaces, metrizability
54H99Connections of general topology with other structures
46B10Duality and reflexivity in normed spaces
46B20Geometry and structure of normed linear spaces
References:
[1]Alegre, C.: Projective limits of paratopological vector spaces, Bull. belg. Math. soc. Simon stevin 12, No. 1, 83-93 (2005) · Zbl 1089.46003 · doi:euclid:bbms/1113318132
[2]Alegre, C.; Ferrer, J.; Gregori, V.: Quasi-uniformities on real vector spaces, Indian J. Pure appl. Math. 28, 929-937 (1997) · Zbl 0917.46003
[3]Alegre, C.; Ferrer, J.; Gregori, V.: On the Hahn – Banach theorem in certain linear quasi-uniform structures, Acta math. Hungar. 82, 315-320 (1999) · Zbl 0930.46004 · doi:10.1023/A:1006692309917
[4]Alegre, C.; Ferrando, I.; García-Raffi, L. M.; Sánchez-Pérez, E. A.: Compactness in asymmetric normed spaces, Topology appl. 155, 527-539 (2008) · Zbl 1142.46004 · doi:10.1016/j.topol.2007.11.004
[5]Alimov, A.: On the structure of the complements of Chebyshev sets, Funct. anal. Appl. 35, 176-182 (2001) · Zbl 1099.41501 · doi:10.1023/A:1012370610709
[6]Beauzamy, B.: Introduction to Banach spaces and their geometry, North-holland math. Stud. (1985)
[7]Ferrer, J.; Gregori, V.; Alegre, A.: Quasi-uniform structures in linear lattices, Rocky mountain J. Math. 23, 877-884 (1993) · Zbl 0803.46007 · doi:10.1216/rmjm/1181072529
[8]Fletcher, P.; Lindgren, W. F.: Quasi-uniform spaces, (1982)
[9]García-Raffi, L. M.; Romaguera, S.; Sánchez-Pérez, E. A.: Extensions of asymmetric norms to linear spaces, Rend. istit. Mat. univ. Trieste 33, 113-125 (2001) · Zbl 1018.46015
[10]García-Raffi, L. M.; Romaguera, S.; Sánchez-Pérez, E. A.: The bicompletion of an asymmetric normed linear space, Acta math. Hungar. 97, No. 3, 183-191 (2002) · Zbl 1012.54031 · doi:10.1023/A:1020823326919
[11]García-Raffi, L. M.; Romaguera, S.; Sánchez-Pérez, E. A.: Sequence spaces and asymmetric norms in the theory of computational complexity, Math. comput. Modelling 36, 1-11 (2002) · Zbl 1063.68057 · doi:10.1016/S0895-7177(02)00100-0
[12]García-Raffi, L. M.; Romaguera, S.; Sánchez-Pérez, E. A.: On Hausdorff asymmetric normed linear spaces, Houston J. Math. 29, 717-728 (2003) · Zbl 1131.46300 · doi:http://www.math.uh.edu/~hjm/restricted/pdf29(3)/12raffi.pdf
[13]García-Raffi, L. M.; Romaguera, S.; Sánchez-Pérez, E. A.: Weak topologies on asymmetric normed linear spaces and non-asymptotic criteria in the theory of complexity analysis of algorithms, J. anal. Appl. 2, 125-138 (2004) · Zbl 1067.46032
[14]García-Raffi, L. M.; Romaguera, S.; Sánchez-Pérez, E. A.: The dual space of an asymmetric normed linear space, Quaest. math. 26, 83-96 (2003) · Zbl 1043.46021 · doi:10.2989/16073600309486046
[15]García-Raffi, L. M.; Sánchez-Pérez, E. A.: Asymmetric norms and optimal distance points in linear spaces, Topology appl. 155, 1410-1419 (2008) · Zbl 1160.46015 · doi:10.1016/j.topol.2008.04.002
[16]Holmes, R.: Geometric functional analysis, (1975)
[17]Jameson, G. J. O.: Topology and normed spaces, (1974)
[18]Kolmogorov, A. N.; Fomin, S. V.: Introductory real analysis, (1975)
[19]Künzi, H. P. A.: Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology, Handbook of the history of general topology, vol. 3 3, 853-968 (2001) · Zbl 1002.54002
[20]Lindenstrauss, J.; Tzafriri, L.: Classical Banach spaces I and II, (1996)
[21]Mustăta, C.: Extensions of semi-Lipschitz functions on quasi-metric spaces, Ann. numer. Theory approx. 30, 61-67 (2001) · Zbl 1010.46004
[22]Mustăta, C.: On the extremal semi-Lipschitz functions, Ann. numer. Theory approx. 31, 103-108 (2002)
[23]Romaguera, S.; Sanchis, M.: Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. approx. Theory 103, 292-301 (2000) · Zbl 0980.41029 · doi:10.1006/jath.1999.3439
[24]Romaguera, S.; Schellekens, M.: Quasi-metric properties of complexity spaces, Topology appl. 98, 311-322 (1999) · Zbl 0941.54028 · doi:10.1016/S0166-8641(98)00102-3
[25]Rudin, W.: Functional analysis, (1973) · Zbl 0253.46001
[26]Schellekens, M.: The smyth completion: A common foundation for denotational semantics and complexity analysis, Electron. notes theor. Comput. sci. 1, 211-232 (1995) · Zbl 0910.68135 · doi:http://www.elsevier.com/cas/tree/store/tcs/free/noncas/pc/volume1.htm#schellekens
[27]Wojtaszczyk, P.: Banach spaces for analysts, (1991)