Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE GEOMETRY OF 𝓛s(3𝒍2∞)

  • Kim, Sung Guen (Department of Mathematics Kyungpook National University)
  • Received : 2017.01.17
  • Accepted : 2017.03.28
  • Published : 2017.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

We classify the extreme, exposed and smooth symmetric 3-linear forms of the unit ball of ${\mathcal{L}}_s(^3l^2_{\infty})$, respectively.

Keywords

References

  1. R. M. Aron, Y. S. Choi, S. G. Kim, and M. Maestre, Local properties of polynomials on a Banach space, Illinois J. Math. 45 (2001), no. 1, 25-39.
  2. W. Cavalcante and D. Pellegrino, Geometry of the closed unit ball of the space of bilinear forms on $l^2_\infty$ , arXiv:1603.01535v2.
  3. Y. S. Choi, H. Ki, and S. G. Kim, Extreme polynomials and multilinear forms on $l_1$, J. Math. Anal. Appl. 228 (1998), no. 2, 467-482. https://doi.org/10.1006/jmaa.1998.6161
  4. Y. S. Choi and S. G. Kim, The unit ball of P($^2l^2_2$), Arch. Math. (Basel) 71 (1998), no. 6, 472-480. https://doi.org/10.1007/s000130050292
  5. Y. S. Choi and S. G. Kim, Extreme polynomials on $c_0$, Indian J. Pure Appl. Math. 29 (1998), no. 10, 983-989.
  6. Y. S. Choi and S. G. Kim, Smooth points of the unit ball of the space P($^2l_1$), Results Math. 36 (1999), no. 1-2, 26-33. https://doi.org/10.1007/BF03322099
  7. Y. S. Choi and S. G. Kim, Exposed points of the unit balls of the spaces P($^2l^2_p$ ) (p = 1, 2, $\infty$), Indian J. Pure Appl. Math. 35 (2004), no. 1, 37-41.
  8. S. Dineen, Complex Analysis on In nite Dimensional Spaces, Springer-Verlag, London, 1999.
  9. S. Dineen, Extreme integral polynomials on a complex Banach space, Math. Scand. 92 (2003), no. 1, 129-140. https://doi.org/10.7146/math.scand.a-14397
  10. D. Diniz, G. A. Munoz-Fernandez, D. Pellegrino, and J. B. Seoane-Sepulveda, Lower bounds for the constants in the Bohnenblust-Hill inequality: the case of real scalars, Proc. Amer. Math. Soc. 142 (2014), no. 2, 575-580. https://doi.org/10.1090/S0002-9939-2013-11791-0
  11. B. C. Grecu, Geometry of 2-homogeneous polynomials on lp spaces, 1 < p < $\infty$, J. Math. Anal. Appl. 273 (2002), no. 2, 262-282. https://doi.org/10.1016/S0022-247X(02)00217-2
  12. B. C. Grecu, G. A. Munoz-Fernandez, and J. B. Seoane-Sepulveda, Unconditional con- stants and polynomial inequalities, J. Approx. Theory 161 (2009), no. 2, 706-722. https://doi.org/10.1016/j.jat.2008.12.001
  13. S. G. Kim, Exposed 2-homogeneous polynomials on P($^2l^2_p$) ($1 \leq p \leq \infty$), Math. Proc. R. Ir. Acad. 107A (2007), no. 2, 123-129.
  14. S. G. Kim, The unit ball of $L_s(^2l^2_\infty)$, Extracta Math. 24 (2009), no. 1, 17-29.
  15. S. G. Kim, The unit ball of P($^2d_*(1, w)^2$), Math. Proc. Royal Irish Acad. 111A (2011), no. 2, 79-94.
  16. S. G. Kim, The unit ball of $L_s(^2d_*(1, w)^2)$, Kyungpook Math. J. 53 (2013), no. 2, 295-306. https://doi.org/10.5666/KMJ.2013.53.2.295
  17. S. G. Kim, Smooth polynomials of P($^2d_*(1, w)^2$), Math. Proc. Royal Irish Acad. 113A (2013), no. 1, 45-58.
  18. S. G. Kim, Extreme bilinear forms of L($2d_*(1;w)^2$), Kyungpook Math. J. 53 (2013), no. 4, 625-638. https://doi.org/10.5666/KMJ.2013.53.4.625
  19. S. G. Kim, Exposed symmetric bilinear forms of $L_s(2d_*(1, w)^2)$, Kyungpook Math. J. 54 (2014), no. 3, 341-347. https://doi.org/10.5666/KMJ.2014.54.3.341
  20. S. G. Kim, Exposed bilinear forms of L($2d_*(1, w)^2$), Kyungpook Math. J. 55 (2015), no. 1, 119-126. https://doi.org/10.5666/KMJ.2015.55.1.119
  21. S. G. Kim, Exposed 2-homogeneous polynomials on the 2-dimensional real predual of Lorentz sequence space, Mediterr. J. Math. 13 (2016), no. 5, 2827-2839. https://doi.org/10.1007/s00009-015-0658-4
  22. S. G. Kim, The geometry of L($^2l^2_\infty$ ), to appear in Kyungpook Math. J.
  23. S. G. Kim, The unit ball of $Ls(^2l^3_\infty)$, to appear in Comment. Math. Prace Mat. 57 (2017).
  24. S. G. Kim and S. H. Lee, Exposed 2-homogeneous polynomials on Hilbert spaces, Proc. Amer. Math. Soc. 131 (2003), no. 2, 449-453. https://doi.org/10.1090/S0002-9939-02-06544-9
  25. J. Lee and K. S. Rim, Properties of symmetric matrices, J. Math. Anal. Appl. 305 (2005), 219-226. https://doi.org/10.1016/j.jmaa.2004.11.011
  26. G. A. Munoz-Fernandez, S. Revesz, and J. B. Seoane-Sepulveda, Geometry of homoge- neous polynomials on non symmetric convex bodies, Math. Scand. 105 (2009), no. 1, 147-160. https://doi.org/10.7146/math.scand.a-15111
  27. G. A. Munoz-Fernandez and J. B. Seoane-Sepulveda, Geometry of Banach spaces of trinomials, J. Math. Anal. Appl. 340 (2008), no. 2, 1069-1087. https://doi.org/10.1016/j.jmaa.2007.09.010
  28. R. A. Ryan and B. Turett, Geometry of spaces of polynomials, J. Math. Anal. Appl. 221 (1998), no. 2, 698-711. https://doi.org/10.1006/jmaa.1998.5942