On the calculation of the Dunkl-Williams constant of normed linear spaces

Hiroyasu Mizuguchi; Kichi-Suke Saito; Ryotaro Tanaka

Open Mathematics (2013)

  • Volume: 11, Issue: 7, page 1212-1227
  • ISSN: 2391-5455

Abstract

top
Recently, Jiménez-Melado et al. [Jiménez-Melado A., Llorens-Fuster E., Mazcuñán-Navarro E.M., The Dunkl-Williams constant, convexity, smoothness and normal structure, J. Math. Anal. Appl., 2008, 342(1), 298–310] defined the Dunkl-Williams constant DW(X) of a normed linear space X. In this paper we present some characterizations of this constant. As an application, we calculate DW(ℓ2-ℓ∞) in the Day-James space ℓ2-ℓ∞.

How to cite

top

Hiroyasu Mizuguchi, Kichi-Suke Saito, and Ryotaro Tanaka. "On the calculation of the Dunkl-Williams constant of normed linear spaces." Open Mathematics 11.7 (2013): 1212-1227. <http://eudml.org/doc/269422>.

@article{HiroyasuMizuguchi2013,
abstract = {Recently, Jiménez-Melado et al. [Jiménez-Melado A., Llorens-Fuster E., Mazcuñán-Navarro E.M., The Dunkl-Williams constant, convexity, smoothness and normal structure, J. Math. Anal. Appl., 2008, 342(1), 298–310] defined the Dunkl-Williams constant DW(X) of a normed linear space X. In this paper we present some characterizations of this constant. As an application, we calculate DW(ℓ2-ℓ∞) in the Day-James space ℓ2-ℓ∞.},
author = {Hiroyasu Mizuguchi, Kichi-Suke Saito, Ryotaro Tanaka},
journal = {Open Mathematics},
keywords = {Dunkl-Williams constant; Birkhoff-James orthogonality; norming functional},
language = {eng},
number = {7},
pages = {1212-1227},
title = {On the calculation of the Dunkl-Williams constant of normed linear spaces},
url = {http://eudml.org/doc/269422},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Hiroyasu Mizuguchi
AU - Kichi-Suke Saito
AU - Ryotaro Tanaka
TI - On the calculation of the Dunkl-Williams constant of normed linear spaces
JO - Open Mathematics
PY - 2013
VL - 11
IS - 7
SP - 1212
EP - 1227
AB - Recently, Jiménez-Melado et al. [Jiménez-Melado A., Llorens-Fuster E., Mazcuñán-Navarro E.M., The Dunkl-Williams constant, convexity, smoothness and normal structure, J. Math. Anal. Appl., 2008, 342(1), 298–310] defined the Dunkl-Williams constant DW(X) of a normed linear space X. In this paper we present some characterizations of this constant. As an application, we calculate DW(ℓ2-ℓ∞) in the Day-James space ℓ2-ℓ∞.
LA - eng
KW - Dunkl-Williams constant; Birkhoff-James orthogonality; norming functional
UR - http://eudml.org/doc/269422
ER -

References

top
  1. [1] Alonso J., Some properties of Birkhoff and isosceles orthogonality in normed linear spaces, In: Inner Product Spaces and Applications, Pitman Res. Notes Math. Ser., 376, Longman, Harlow, 1997, 1–11 Zbl0898.46019
  2. [2] Alonso J., Martini H., Mustafaev Z., On orthogonal chords in normed planes, Rocky Mountain J. Math., 2011, 41(1), 23–35 http://dx.doi.org/10.1216/RMJ-2011-41-1-23[Crossref] Zbl1214.52001
  3. [3] Al-Rashed A.M., Norm inequalities and characterizations of inner product spaces, J. Math. Anal. Appl., 1993, 176(2), 587–593 http://dx.doi.org/10.1006/jmaa.1993.1233[Crossref] 
  4. [4] Birkhoff G., Orthogonality in linear metric spaces, Duke Math. J., 1935, 1(2), 169–172 http://dx.doi.org/10.1215/S0012-7094-35-00115-6[Crossref] Zbl0012.30604
  5. [5] Dadipour F., Fujii M., Moslehian M.S., Dunkl-Williams inequality for operators associated with p-angular distance, Nihonkai Math. J., 2010, 21(1), 11–20 Zbl1225.47020
  6. [6] Dadipour F., Moslehian M.S., An approach to operator Dunkl-Williams inequalities, Publ. Math. Debrecen, 2011, 79(1-2), 109–118 http://dx.doi.org/10.5486/PMD.2011.4936[Crossref][WoS] Zbl1262.47026
  7. [7] Day M.M., Some characterizations of inner-product spaces, Trans. Amer. Math. Soc., 1947, 62, 320–337 http://dx.doi.org/10.1090/S0002-9947-1947-0022312-9[Crossref] Zbl0034.21703
  8. [8] Dunkl C.F., Williams K.S., A simple norm inequality, Amer. Math. Monthly, 1964, 71(1), 53–54 http://dx.doi.org/10.2307/2311304[Crossref] Zbl0178.16202
  9. [9] James R.C., Orthogonality in normed linear spaces, Duke Math. J., 1945, 12(2), 291–302 http://dx.doi.org/10.1215/S0012-7094-45-01223-3[Crossref] Zbl0060.26202
  10. [10] James R.C., Inner product in normed linear spaces, Bull. Amer. Math. Soc., 1947, 53(6), 559–566 http://dx.doi.org/10.1090/S0002-9904-1947-08831-5[Crossref] Zbl0041.43701
  11. [11] James R.C., Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc., 1947, 61(2), 265–292 http://dx.doi.org/10.1090/S0002-9947-1947-0021241-4[Crossref] 
  12. [12] Jiménez-Melado A., Llorens-Fuster E., Mazcuñán-Navarro E.M., The Dunkl-Williams constant, convexity, smoothness and normal structure, J. Math. Anal. Appl., 2008, 342(1), 298–310 http://dx.doi.org/10.1016/j.jmaa.2007.11.045[Crossref][WoS] Zbl1151.46009
  13. [13] Kato M., Saito K.-S., Tamura T., Sharp triangle inequality and its reverse in Banach spaces, Math. Inequal. Appl., 2007, 10(2), 451–460 Zbl1121.46019
  14. [14] Kirk W.A., Smiley M.F., Another characterization of inner product spaces, Amer. Math. Monthly, 1964, 71(8), 890–891 http://dx.doi.org/10.2307/2312400[Crossref] 
  15. [15] Megginson R.E., An Introduction to Banach Space Theory, Grad. Texts in Math., 183, Springer, New York, 1998 http://dx.doi.org/10.1007/978-1-4612-0603-3[Crossref] 
  16. [16] Moslehian M.S., Dadipour F., Rajic R., Maric A., A glimpse at the Dunkl-Williams inequality, Banach J. Math. Anal., 2011, 5(2), 138–151 Zbl1225.47022
  17. [17] Nilsrakoo W., Saejung S., The James constant of normalized norms on R2, J. Inequal. Appl., 2006, #26265 Zbl1104.46007
  18. [18] Pečaric J., Rajic R., Inequalities of the Dunkl-Williams type for absolute value operators, J. Math. Inequal., 2010, 4(1), 1–10 http://dx.doi.org/10.7153/jmi-04-01[Crossref] Zbl1186.26020
  19. [19] Saito K.-S., Tominaga M., A Dunkl-Williams type inequality for absolute value operators, Linear Algebra Appl., 2010, 432(12), 3258–3264 http://dx.doi.org/10.1016/j.laa.2010.01.016[WoS][Crossref] Zbl1195.26044

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.