Abstract

Based on an idea of Ivan Singer we introduce a new concept of an angle in real normed vector spaces which generalizes the Euclidean angle of Hilbert spaces. With this angle we can describe elements of each finite dimensional real normed space by polar coordinates.

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Normed vector space
  • generalized angle
  • polar coordinates

Mathematics Subject Classification

46B20, 52A10

References

  1. D. Amir: Characterizations of Inner Product Spaces, Birkhäuser, Basel (1986).https://doi.org/10.1007/978-3-0348-5487-0Google Scholar
  2. C. Benitez: Orthogonality in normed linear spaces: a classification of the different concepts and some open problems, Rev. Mat. Univ. Complutense Madr. 2, Suppl. (1989) 53–57.https://doi.org/10.5209/rev_REMA.1989.v2.18066Google Scholar
  3. C. Diminnie, E. Andalafte, R. Freese: Angles in normed linear spaces and a characterization of real inner product spaces, Math. Nachr. 129 (1986) 197–204.https://doi.org/10.1002/mana.19861290118Google Scholar
  4. C. Diminnie, E. Andalafte, R. Freese: Generalized angles and a characterization of inner product spaces, Houston J. Math. 14(4) (1988) 457–480.Google Scholar
  5. S. Gudder, D. Strawther: Orthogonally additive and orthogonally increasing functions on vector spaces, Pac. J. Math. 58(2) (1975) 427–436.https://doi.org/10.2140/pjm.1975.58.427Google Scholar
  6. H. Gunawan, J. Lindiarni, O. Neswan: P -, I-, g-, and D-angles in normed spaces, ITB J. Sci. 40A(1) (2008) 24–32.https://doi.org/10.5614/itbj.sci.2008.40.1.3Google Scholar
  7. H. Martini, K. J. Swanepoel, G. Weiss: The geometry of Minkowski spaces - a survey. I., Expo. Math. 19(2) (2001) 97–142.https://doi.org/10.1016/S0723-0869(01)80025-6Google Scholar
  8. R. Meise, D. Vogt: Introduction to Functional Analysis, Oxford University Press, Oxford (1997).https://doi.org/10.1093/oso/9780198514855.001.0001Google Scholar
  9. K. Menger: Some applications of point-set methods, Ann. Math. 32 (1931) 739–750.https://doi.org/10.2307/1968317Google Scholar
  10. P. M. Miličić: Sur le g-angle dans un espace norme, Mat. Vesn. 45 (1993) 43–48.Google Scholar
  11. P. M. Miličić: Characterizations of convexities of normed spaces by means of g-angles, Mat. Vesn. 54 (2002) 37–44.Google Scholar
  12. P. M. Miličić: On the B-angle and g-angle in normed spaces, JIPAM, J. Inequal. Pure Appl. Math. 8(4), Paper 99 (2007), 18 p.Google Scholar
  13. P. M. Miličić: The Thy-angle and g-angle in a quasi-inner product space, Math. Morav. 15(2) (2011) 41–46.https://doi.org/10.5937/MatMor1102041MGoogle Scholar
  14. W. Rudin: Functional Analysis, MacGraw-Hill, New York (1991).Google Scholar
  15. I. Singer: Unghiuri abstracte şi funcţii trigonometrice în spaţii Banach, Acad. Republ. Popul. Romîne, Bul. Şti., Secţ. Şti. Mat. Fiz. 9 (1957) 29–42.Google Scholar
  16. V. W. Thürey: Angles and polar coordinates in real normed spaces, arXiv:0902.2731 [math.FA] (2009).Google Scholar
  17. V. W. Thürey: Angles and a classification of normed spaces, Ann. Funct. Anal. AFA 4(1) (2013) 114–137.https://doi.org/10.15352/afa/1399899841Google Scholar
  18. J. E. Valentine, C. Martin: Angles in metric and normed linear spaces, Colloq. Math. 34 (1976) 209–217.https://doi.org/10.4064/cm-34-2-209-217Google Scholar
  19. J. E. Valentine, S. G. Wayment: Wilson angles in linear normed spaces, Pac. J. Math. 36(1) (1971) 239–243.https://doi.org/10.2140/pjm.1971.36.239Google Scholar

Suggested citation

V. W. Thürey. “A Generalization of the Euclidean Angle.” Journal of Convex Analysis 20 (2013), No. 4, 1025–1042. https://doi.org/10.68381/jca20056

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.