A universal modulus for normed spaces

Carlos Benítez; Krzysztof Przesławski; David Yost

Studia Mathematica (1998)

  • Volume: 127, Issue: 1, page 21-46
  • ISSN: 0039-3223

Abstract

top
We define a handy new modulus for normed spaces. More precisely, given any normed space X, we define in a canonical way a function ξ:[0,1)→ ℝ which depends only on the two-dimensional subspaces of X. We show that this function is strictly increasing and convex, and that its behaviour is intimately connected with the geometry of X. In particular, ξ tells us whether or not X is uniformly smooth, uniformly convex, uniformly non-square or an inner product space.

How to cite

top

Benítez, Carlos, Przesławski, Krzysztof, and Yost, David. "A universal modulus for normed spaces." Studia Mathematica 127.1 (1998): 21-46. <http://eudml.org/doc/216458>.

@article{Benítez1998,
abstract = {We define a handy new modulus for normed spaces. More precisely, given any normed space X, we define in a canonical way a function ξ:[0,1)→ ℝ which depends only on the two-dimensional subspaces of X. We show that this function is strictly increasing and convex, and that its behaviour is intimately connected with the geometry of X. In particular, ξ tells us whether or not X is uniformly smooth, uniformly convex, uniformly non-square or an inner product space.},
author = {Benítez, Carlos, Przesławski, Krzysztof, Yost, David},
journal = {Studia Mathematica},
keywords = {universal modulus; strictly increasing and convex; geometry; uniformly smooth; uniformly convex; uniformly non-square; inner product space},
language = {eng},
number = {1},
pages = {21-46},
title = {A universal modulus for normed spaces},
url = {http://eudml.org/doc/216458},
volume = {127},
year = {1998},
}

TY - JOUR
AU - Benítez, Carlos
AU - Przesławski, Krzysztof
AU - Yost, David
TI - A universal modulus for normed spaces
JO - Studia Mathematica
PY - 1998
VL - 127
IS - 1
SP - 21
EP - 46
AB - We define a handy new modulus for normed spaces. More precisely, given any normed space X, we define in a canonical way a function ξ:[0,1)→ ℝ which depends only on the two-dimensional subspaces of X. We show that this function is strictly increasing and convex, and that its behaviour is intimately connected with the geometry of X. In particular, ξ tells us whether or not X is uniformly smooth, uniformly convex, uniformly non-square or an inner product space.
LA - eng
KW - universal modulus; strictly increasing and convex; geometry; uniformly smooth; uniformly convex; uniformly non-square; inner product space
UR - http://eudml.org/doc/216458
ER -

References

top
  1. [1] J. Alonso and C. Benítez, Some characteristic and non-characteristic properties of inner product spaces, J. Approx. Theory 55 (1988), 318-323. Zbl0675.41047
  2. [2] J. Alonso and A. Ullán, Moduli in normed linear spaces and characterization of inner product spaces, Arch. Math. (Basel) 59 (1992), 487-495. Zbl0760.46014
  3. [3] D. Amir, Characterizations of Inner Product Spaces, Birkhäuser, Basel, 1986. Zbl0617.46030
  4. [4] C. Benítez and M. del Río, Characterization of inner product spaces through rectangle and square inequalities, Rev. Roumaine Math. Pures Appl. 29 (1984), 543-546. Zbl0556.46013
  5. [5] C. Benítez and Y. Sarantopoulos, Characterization of real inner product spaces by means of symmetric bilinear forms, J. Math. Anal. Appl. 180 (1993), 207-220. Zbl0791.46015
  6. [6] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935), 169-172. Zbl0012.30604
  7. [7] F. Cabello Sánchez and J. M. F. Castillo, Isometries of finite dimensional normed spaces, Extracta Math. 10 (1995), 146-151. 
  8. [8] M. M. Day, Polygons circumscribed about closed convex curves, Trans. Amer. Math. Soc. 62 (1947), 315-319. Zbl0034.25301
  9. [9] M. M. Day, Some characterizations of inner-product spaces, ibid., 320-337. Zbl0034.21703
  10. [10] M. del Río and C. Benítez, The rectangular constant for two-dimensional normed spaces, J. Approx. Theory 19 (1977), 15-21. Zbl0343.46018
  11. [11] J. Gao and K. S. Lau, On two classes of Banach spaces with uniform normal structure, Studia Math. 99 (1991), 41-56. Zbl0757.46023
  12. [12] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, 1990. Zbl0708.47031
  13. [13] V. I. Gurariĭ, On differential properties of the modulus of convexity of Banach spaces, Mat. Issled. 2 (1967), 141-148 (in Russian). 
  14. [14] R. E. Harrell and L. A. Karlovitz, Girths and flat Banach spaces, Bull. Amer. Math. Soc. 76 (1970) 1288-1291. Zbl0212.14303
  15. [15] J.-B. Hiriart-Urruty, A short proof of the variational principle for approximate solutions of a minimization problem, Amer. Math. Monthly 90 (1983), 206-207. 
  16. [16] R. C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265-292. 
  17. [17] R. C. James, Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), 540-550. Zbl0132.08902
  18. [18] J. Joly, Caractérisations d'espaces hilbertiens au moyen de la constante rectangle, J. Approx. Theory 2 (1969), 301-311. Zbl0177.40701
  19. [19] M. I. Kadets, Proof of the topological equivalence of all separable infinite-dimensional Banach spaces, Functional Anal. Appl. 1 (1967), 53-62 (= Funktsional. Anal. i Prilozhen. 1 (1967), 61-70). 
  20. [20] M. A. Khamsi, Étude de la propriété du point fixe dans les espaces de Banach et les espaces métriques, thesis, Univ. Paris VI, 1987. 
  21. [21] V. I. Liokumovich, Existence of B-spaces with a non-convex modulus of convexity, Izv. Vyssh. Uchebn. Zaved. Mat. 12 (1973), 43-49 (in Russian). Zbl0274.46014
  22. [22] V. D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer, Berlin, 1986. Zbl0606.46013
  23. [23] S. Prus, A remark on a theorem of Turett, Bull. Polish Acad. Sci. Math. 36 (1988), 225-227. Zbl0767.46009
  24. [24] K. Przesławski and D. Yost, Lipschitz retracts, selectors and extensions, Michigan Math. J. 42 (1995), 555-571. Zbl0920.54019
  25. [25] S. Rolewicz, On drop property, Studia Math. 85 (1986), 27-35. Zbl0642.46011
  26. [26] J. J. Schäffer, Inner diameter, perimeter, and girth of spheres, Math. Ann. 173 (1967), 59-79 and 79-82. Zbl0152.12405
  27. [27] B. Sims, unpublished seminar notes, Kent State Univ., Kent, Ohio, 1986. 
  28. [28] C. Zanco and A. Zucchi, Moduli of rotundity and smoothness for convex bodies, Boll. Un. Mat. Ital. B (7) 7 (1993), 833-855. Zbl0804.52001

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.