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Mat. Zametki, 2001, Volume 69, Issue 3, Pages 329–337 (Mi mz506)  

This article is cited in 14 scientific papers (total in 14 papers)

The Banach–Mazur Theorem for Spaces with Asymmetric Norm

P. A. Borodin

M. V. Lomonosov Moscow State University

Abstract: We establish an analog of the Banach–Mazur theorem for real separable linear spaces with asymmetric norm: every such space can be linearly and isometrically embedded in the space of continuous functions $f$ on the interval $[0,1]$ equipped with the asymmetric norm $\|f|=\max\{f(t)\colon t\in[0,1]\}$. This assertion is used to obtain nontrivial representations of an arbitrary convex closed body $M\subset\mathbb R^n$ , an arbitrary compact set $K\subset\mathbb R^n$, and an arbitrary continuous function $F\colon K\to\mathbb R$.

DOI: https://doi.org/10.4213/mzm506

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English version:
Mathematical Notes, 2001, 69:3, 298–305

Bibliographic databases:

UDC: 517.982
Received: 24.01.2000

Citation: P. A. Borodin, “The Banach–Mazur Theorem for Spaces with Asymmetric Norm”, Mat. Zametki, 69:3 (2001), 329–337; Math. Notes, 69:3 (2001), 298–305

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. R. Alimov, “The Banach–Mazur theorem for spaces with an asymmetric distance”, Russian Math. Surveys, 58:2 (2003), 367–369  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. A. R. Alimov, “Convexity of Chebyshev Sets Contained in a Subspace”, Math. Notes, 78:1 (2005), 3–13  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Raffi, LG, “Asymmetric norms and optimal distance points in linear spaces”, Topology and Its Applications, 155:13 (2008), 1410  crossref  mathscinet  zmath  isi  scopus  scopus
    4. Mustata, C, “On the Approximation of the Global Extremum of a Semi-Lipschitz Function”, Mediterranean Journal of Mathematics, 6:2 (2009), 169  crossref  mathscinet  zmath  isi  scopus  scopus
    5. P. A. Borodin, “On the convexity of $N$-Chebyshev sets”, Izv. Math., 75:5 (2011), 889–914  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. Balan G., Richards D., Luke S., “Long-Term Fairness with Bounded Worst-Case Losses”, Auton. Agents Multi-Agent Syst., 22:1 (2011), 43–63  crossref  mathscinet  isi  elib  scopus  scopus
    7. Alegre C., Romaguera S., Veeramani P., “The Uniform Boundedness Theorem in Asymmetric Normed Spaces”, Abstract Appl. Anal., 2012, 809626  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    8. Kocinac L.D.R., Kuenzi H.-P.A., “Selection Properties of Uniform and Related Structures”, Topology Appl., 160:18, SI (2013), 2495–2504  crossref  mathscinet  zmath  isi  scopus  scopus
    9. Belavkin R.V., “Asymmetric Topologies on Statistical Manifolds”, Geometric Science of Information, Lecture Notes in Computer Science, 9389, eds. Nielsen F., Barbaresco F., Springer Int Publishing Ag, 2015, 203–210  crossref  mathscinet  zmath  isi  scopus
    10. Jonard-Perez N., Sanchez-Perez E.A., “Compact convex sets in 2-dimensional asymmetric normed lattices”, Quaest. Math., 39:1 (2016), 73–82  crossref  mathscinet  isi  elib  scopus
    11. Stonyakin F.S., “Subdifferential Calculus in Abstract Convex Cones”, 2017 Constructive Nonsmooth Analysis and Related Topics (Dedicated to the Memory of V. F. Demyanov) (CNSA), ed. Polyakova L., IEEE, 2017, 316–319  isi
    12. Jonard-Perez N., Sanchez-Perez E.A., “Local Compactness in Right Bounded Asymmetric Normed Spaces”, Quaest. Math., 41:4 (2018), 549–563  crossref  mathscinet  zmath  isi  scopus
    13. F. S. Stonyakin, “A Sublinear Analog of the Banach–Mazur Theorem in Separable Convex Cones with Norm”, Math. Notes, 104:1 (2018), 111–120  mathnet  crossref  crossref  isi  elib
    14. F. S. Stonyakin, “Hahn–Banach type theorems on functional separation for convex ordered normed cones”, Eurasian Math. J., 10:1 (2019), 59–79  mathnet  crossref
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