Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CHARACTERIZATIONS OF GEOMETRICAL PROPERTIES OF BANACH SPACES USING ψ-DIRECT SUMS

  • Zhang, Zhihua (School of Mathematical Sciences University of Electronic Science and Technology of China) ;
  • Shu, Lan (School of Mathematical Sciences University of Electronic Science and Technology of China) ;
  • Zheng, Jun (School of Mathematics and Statistics Lanzhou University) ;
  • Yang, Yuling (School of Mathematical Sciences University of Electronic Science and Technology of China)
  • Received : 2011.09.08
  • Published : 2013.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let X be a Banach space and ${\psi}$ a continuous convex function on ${\Delta}_{K+1}$ satisfying certain conditions. Let $(X{\bigoplus}X{\bigoplus}{\cdots}{\bigoplus}X)_{\psi}$ be the ${\psi}$-direct sum of X. In this paper, we characterize the K strict convexity, K uniform convexity and uniform non-$l^N_1$-ness of Banach spaces using ${\psi}$-direct sums.

Keywords

Acknowledgement

Supported by : Nation Natural Science Foundation of China

References

  1. B. Beauzamy, Introduction to Banach Spaces and Their Geometry, 2nd ed., North Holland, 1985.
  2. F. F. Bonsall and J. Duncan, Numerical Ranges. II, London Mathematical Society Lecture Notes Series, No. 10. Cambridge University Press, New York-London, 1973.
  3. S. Dhompongsa, A. Kaewcharoen, and A. Kaewkhao, Fixed point property of direct sums, Nonlinear Anal. 63 (2005), 2177-2188. https://doi.org/10.1016/j.na.2005.02.020
  4. S. Dhompongsa, A. Kaewkhao, and S. Saejung, Uniform smoothness and U-convexity of ${\psi}$-direct sums, J. Nonlinear Convex Anal. 6 (2005), no. 2, 327-338.
  5. P. N. Dowling and B. Turett, Complex strict convexity of absolute norms on Cn and direct sums of Banach spaces, J. Math. Anal. Appl. 323 (2006), no. 2, 930-937. https://doi.org/10.1016/j.jmaa.2005.11.007
  6. M. Kato, K.-S. Saito, and T. Tamura, On ${\psi}$-direct sums of Banach spaces and convexity, J. Aust. Math. Soc. 75 (2003), no. 3, 413-422. https://doi.org/10.1017/S1446788700008193
  7. M. Kato, K.-S. Saito, and T. Tamura, Uniform non-squareness of ${\psi}$-direct sums of Banach spaces $X{\bigoplus}_{\psi}Y$, Math. Inequal. Appl. 7 (2004), no. 3, 429-437.
  8. M. Kato, K.-S. Saito, and T. Tamura, Sharp triangle inequality and its reverse in Banach spaces, Math. Inequal. Appl. 10 (2007), no. 2, 451-460.
  9. M. Kato, K.-S. Saito, and T. Tamura, Uniform non- $l_{1}^{n}$-ness of ${\psi}$-direct sums of Banach spaces, J. Nonlinear Convex Anal. 11 (2010), no. 1, 13-33.
  10. R. E. Megginson, An Introduction to Banach Spaces Theory, Springer, 1998.
  11. K.-I. Mitani and K.-S. Saito, A note on geometrical properties of Banach spaces using ${\psi}$-direct sums, J. Math. Anal. Appl. 327 (2007), no. 2, 898-907. https://doi.org/10.1016/j.jmaa.2006.04.059
  12. K.-S. Saito and M. Kato, Uniform convexity of ${\psi}$-direct sums of Banach spaces, J. Math. Anal. Appl. 277 (2003), no. 1, 1-11. https://doi.org/10.1016/S0022-247X(02)00282-2
  13. K.-S. Saito, M. Kato, and Y. Takahashi, Absolute norms on $\mathbb{C}^{n}$, J. Math. Anal. Appl. 252 (2000), no. 2, 879-905. https://doi.org/10.1006/jmaa.2000.7139
  14. I. Singer, On the set of best approximation of an element in a normed linear space, Rev. Math. Pures Appl. 5 (1960), 383-402.
  15. F. Sullivan, A generalization of uniformly rotund Banach spaces, Can. J. Math. 31 (1979), no. 3, 628-636. https://doi.org/10.4153/CJM-1979-063-9
  16. Y. Takahashi, M. Kato, and K.-S. Saito, Strict convexity of absolute norms on $\mathbb{C}^{2}$ and direct sums of Banach spaces, J. Inequal. Appl. 7 (2002), no. 2, 179-186.
  17. X. T. Yu, E. B. Zang, and Z. Liu, On KUR Banach spaces, J. East China Normal Univ. Nature Science Edition. 1 (1981), 1-8.