DOI QRμ½”λ“œ

DOI QR Code

Exposed Bilinear Forms of 𝓛(2d*(1, w)2)

  • Kim, Sung Guen (Department of Mathematics, Kyungpook National University)
  • Received : 2014.01.02
  • Accepted : 2014.04.09
  • Published : 2015.03.23

Abstract

First we present the explicit formula for the norm of a (continuous) linear functional of $\mathcal{L}(^2d_*(1,w)^2)^*$. Using this formula and results of [16] and [17], we show that every extreme bilinear form of the unit ball of $\mathcal{L}(^2d_*(1,w)^2)$ is exposed.

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), 25-39.
  2. 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
  3. 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
  4. Y. S. Choi and S. G. Kim, Extreme polynomials on $c_0$, Indian J. Pure Appl. Math., 29(1998), 983-989.
  5. 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
  6. 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), 37-41.
  7. S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag, London (1999).
  8. S. Dineen, Extreme integral polynomials on a complex Banach space, Math. Scand., 92(2003), 129-140.
  9. 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
  10. 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
  11. S. G. Kim, Exposed 2-homogeneous polynomials on $P(^2l^2_p)$ $1{\leq}p{\leq}{\infty}$, Math. Proc. Royal Irish Acad., 107(2007), 123-129. https://doi.org/10.3318/PRIA.2007.107.2.123
  12. S. G. Kim, The unit ball of $L_s(^2l^2_{\infty})$, Extracta Math., 24(2009), 17-29.
  13. S. G. Kim, The unit ball of $P^2d_*(1,w)^2$, Math. Proc. Royal Irish Acad., 111(2)(2011), 79-94.
  14. 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
  15. S. G. Kim, Smooth polynomials of $P(^2d_*(1,w)^2)$, Math. Proc. Royal Irish Acad., 113A(1)(2013), 45-58.
  16. 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
  17. S. G. Kim, Exposed symmetric bilinear forms of $L(^2d_*(1,w)^2)$, Kyungpook Math. J., 54(2014), 341-347. https://doi.org/10.5666/KMJ.2014.54.3.341
  18. 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
  19. 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
  20. G. A. Munoz-Fernandez, S. Revesz and J. B. Seoane-Sepulveda, Geometry of homogeneous polynomials on non symmetric convex bodies, Math. Scand., 105(2009), 147-160.
  21. 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
  22. 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

Cited by

  1. Extreme bilinear forms on $$\mathbb {R}^n$$Rn with the supremum norm vol.77, pp.2, 2018, https://doi.org/10.1007/s10998-018-0246-z