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The Unit Ball of $\mathcal{L}_s(^2d_*(1,w)^2)$

Kim, Sung Guen

  • Received : 2012.06.18
  • Accepted : 2012.11.07
  • Published : 2013.06.23

Abstract

First we present the explicit formula for the norm of a symmetric bilinear form on the 2-dimensional real predual of the Lorentz sequence space $d_*(1,w)^2$. Using this formula, we classify the extreme points of the unit ball of $\mathcal{L}_s(^2d_*(1,w)^2)$.

Keywords

extreme symmetric bilinear forms;the 2-dimensional real predual of the Lorentz sequence space

References

  1. R. M. Aron, Y. S. Choi, S. G. Kim and M. Maestre, Local properties of poly-nomials 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, Extreme polynomials on $c_{0}$, Indian J. Pure Appl. Math., 29(1998), 983-989.
  4. Y. S. Choi and S. G. Kim, Smooth points of the unit ball of the space $p(^{2}l_{1})$, Results Math., 36(1999), 26-33. https://doi.org/10.1007/BF03322099
  5. Y. S. Choi and S. G. Kim, Exposed points of the unit balls of the spaces $p(^{2}l_{p}^{2})$ (p = 1, 2,${\infty}$), Indian J. Pure Appl. Math., 35(2004), 37-41.
  6. S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag, London (1999).
  7. 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
  8. B. C. Grecu, Geometry of 2-homogeneous polynomials on $l_{p}$ spaces, 1 < p < ${\infty}$, J. Math. Anal. Appl., 273(2002), 262-282 . https://doi.org/10.1016/S0022-247X(02)00217-2
  9. 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
  10. S. G. Kim, The unit ball of $L_{s}(^{2}l^{2}_{\infty})$, Extracta Math., 24(2009), 17-29.
  11. S. G. Kim, The unit ball of $P(^{2}d_{\ast}(1,w)^{2})$, Math. Proc. Royal Irish Acad., 111A(2011), 79-94.
  12. 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
  13. 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
  14. 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. https://doi.org/10.7146/math.scand.a-15111
  15. 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
  16. 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
  17. Y. S. Choi and S. G. Kim, The unit ball of $p(^{2}l_{2}^{2})$, Arch. Math., (Basel), 71(1998), 472-480. https://doi.org/10.1007/s000130050292
  18. S. G. Kim, Exposed 2-homogeneous polynomials on $p(^{2}l_{p}^{2})$ (1 $\leq$ p $\leq$ ${\infty}$), Math. Proc. Royal Irish Acad., 107A(2007), 123-129.

Cited by

  1. Extreme Bilinear Forms of $\mathcal{L}(^2d_*(1,w)^2)$ vol.53, pp.4, 2013, https://doi.org/10.5666/KMJ.2013.53.4.625
  2. Exposed 2-Homogeneous Polynomials on the two-Dimensional Real Predual of Lorentz Sequence Space vol.13, pp.5, 2016, https://doi.org/10.1007/s00009-015-0658-4
  3. Exposed Bilinear Forms of 𝓛(2d*(1, w)2) vol.55, pp.1, 2015, https://doi.org/10.5666/KMJ.2015.55.1.119
  4. Exposed Symmetric Bilinear Forms of 𝓛s(2d*(1, ω)2) vol.54, pp.3, 2014, https://doi.org/10.5666/KMJ.2014.54.3.341
  5. POLARIZATION AND UNCONDITIONAL CONSTANTS OF 𝓟(2d*(1,ω)2) vol.29, pp.3, 2014, https://doi.org/10.4134/CKMS.2014.29.3.421
  6. 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

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)