DOI QR코드

DOI QR Code

The Geometry of Ꮭ(2l2)

  • Kim, Sung Guen (Department of Mathematics, Kyungpook National University)
  • Received : 2015.10.08
  • Accepted : 2016.01.18
  • Published : 2018.03.23

Abstract

We classify the extreme, exposed and smooth bilinear forms of the unit ball of the space of bilinear forms on $l^2_{\infty}$.

Keywords

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

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), 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), 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), 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), 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), 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 $P(^2l_p^2)\;(p=1,\;2,\;{\infty})$, Indian J. Pure Appl. Math., 35(2004), 37-41.
  8. S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag, London (1999).
  9. S. Dineen, Extreme integral polynomials on a complex Banach space, Math. Scand., 92(2003), 129-140. https://doi.org/10.7146/math.scand.a-14397
  10. B. C. Grecu, Geometry of 2-homogeneous polynomials on lp spaces, 1 < p < ${\infty}$, J. Math. Anal. Appl., 273(2002), 262-282. https://doi.org/10.1016/S0022-247X(02)00217-2
  11. B. C. Grecu, G. A. Munoz-Fernandez and J. B. Seoane-Sepulveda,, Unconditional constants and polynomial inequalities, J. Approx. Theory, 161(2009), 706-722. https://doi.org/10.1016/j.jat.2008.12.001
  12. S. G. Kim, Exposed 2-homogeneous polynomials on $P(^2l^2_p)\;(1\;{\leq}\;p\;{\leq}\;{\infty})$, Math. Proc. R. Ir. Acad., 107A(2007), 123-129.
  13. S. G. Kim, The unit ball of $L_s(^2l^2_{\infty})$, Extracta Math., 24(2009), 17-29.
  14. S. G. Kim, The unit ball of $P(^2d_*(1,\;w)^2)$, Math. Proc. R. Ir. Acad., 111A(2011), 79-94.
  15. S. G. Kim, The unit ball of $L_s(^2d_*(1,\;w)^2)$, Kyungpook Math. J., 53(2013), 295-306. https://doi.org/10.5666/KMJ.2013.53.2.295
  16. S. G. Kim, Smooth polynomials of $P(^2d_*(1,\;w)^2)$, Math. Proc. R. Ir. Acad., 113A(2013), 45-58.
  17. S. G. Kim, Extreme bilinear forms of $L(^2d_*(1,\;w)^2)$, Kyungpook Math. J., 53(2013), 625-638. https://doi.org/10.5666/KMJ.2013.53.4.625
  18. S. G. Kim, Exposed symmetric bilinear forms of $L_s(^2d_*(1,\;w)^2)$, Kyungpook Math. J., 54(2014), 341-347. https://doi.org/10.5666/KMJ.2014.54.3.341
  19. S. G. Kim, Exposed bilinear forms of $L(^2d_*(1,\;w)^2)$, Kyungpook Math. J., 55(2015), 119-126. https://doi.org/10.5666/KMJ.2015.55.1.119
  20. S. G. Kim, Exposed 2-homogeneous polynomials on the 2-dimensional real predual of Lorentz sequence space, Mediterr. J. Math., 7(2015), 1-13
  21. S. G. Kim and S.H. Lee, Exposed 2-homogeneous polynomials on Hilbert spaces, Proc. Amer. Math. Soc., 131(2003), 449-453. https://doi.org/10.1090/S0002-9939-02-06544-9
  22. 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
  23. G. A. Munoz-Fernandez, S. Revesz and J. B. Seoane-Sepulveda, Geometry of ho- mogeneous polynomials on non symmetric convex bodies, Math. Scand., 105(2009), 147-160. https://doi.org/10.7146/math.scand.a-15111
  24. G. A. Munoz-Fernandez and J. B. Seoane-Sepulveda, Geometry of Banach spaces of trinomials, J. Math. Anal. Appl., 340(2008), 1069-1087. https://doi.org/10.1016/j.jmaa.2007.09.010
  25. R. A. Ryan and B. Turett Geometry of spaces of polynomials, J. Math. Anal. Appl., 221(1998), 698-711. https://doi.org/10.1006/jmaa.1998.5942