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GEOMETRY OF BILINEAR FORMS ON A NORMED SPACE ℝn

  • Sung Guen Kim (Department of Mathematics Kyungpook National University)
  • Received : 2022.06.02
  • Accepted : 2022.09.26
  • Published : 2023.01.01

Abstract

For every n ≥ 2, let ℝn‖·‖ be Rn with a norm ‖·‖ such that its unit ball has finitely many extreme points more than 2n. We devote to the description of the sets of extreme and exposed points of the closed unit balls of 𝓛(2n‖·‖) and 𝓛𝒮(2n‖·‖), where 𝓛(2n‖·‖) is the space of bilinear forms on ℝn‖·‖, and 𝓛𝒮(2n‖·‖) is the subspace of 𝓛(2n‖·‖) consisting of symmetric bilinear forms. Let 𝓕 = 𝓛(2n‖·‖) or 𝓛𝒮(2n‖·‖). First we classify the extreme and exposed points of the closed unit ball of 𝓕. We also show that every extreme point of the closed unit ball of 𝓕 is exposed. It is shown that ext B𝓛𝒮(2n‖·‖) = ext B𝓛(2n‖·‖) ∩ 𝓛𝒮(2n‖·‖) and exp B𝓛𝒮(2n‖·‖) = exp B𝓛(2n‖·‖) ∩ 𝓛𝒮(2n‖·‖), which expand some results of [18, 23, 28, 29, 35, 38, 40, 41, 43].

Keywords

References

  1. R. M. Aron and M. Klimek, Supremum norms for quadratic polynomials, Arch. Math. (Basel) 76 (2001), no. 1, 73-80. https://doi.org/10.1007/s000130050544
  2. W. V. Cavalcante, D. M. Pellegrino, and E. V. Teixeira, Geometry of multilinear forms, Commun. Contemp. Math. 22 (2020), no. 2, 1950011, 26 pp. https://doi.org/10.1142/S0219199719500111
  3. Y. S. Choi and S. G. Kim, The unit ball of 𝓟(2𝑙22), Arch. Math. (Basel) 71 (1998), no. 6, 472-480. https://doi.org/10.1007/s000130050292
  4. Y. S. Choi and S. G. Kim, Extreme polynomials on c0, Indian J. Pure Appl. Math. 29 (1998), no. 10, 983-989.
  5. Y. S. Choi and S. G. Kim, Smooth points of the unit ball of the space 𝓟(2𝑙1), Results Math. 36 (1999), no. 1-2, 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(2𝑙2p) (p = 1, 2, ∞), Indian J. Pure Appl. Math. 35 (2004), no. 1, 37-41.
  7. Y. S. Choi, S. G. Kim, and H. Ki, Extreme polynomials and multilinear forms on 𝑙1, J. Math. Anal. Appl. 228 (1998), no. 2, 467-482. https://doi.org/10.1006/jmaa.1998.6161
  8. S. Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. https://doi.org/10.1007/978-1-4471-0869-6
  9. J. L. Gamez-Merino, G. A. Munoz-Fernandez, V. M. Sanchez, and J. B. Seoane-Sepulveda, Inequalities for polynomials on the unit square via the Krein-Milman theorem, J. Convex Anal. 20 (2013), no. 1, 125-142.
  10. B. C. Grecu, Geometry of three-homogeneous polynomials on real Hilbert spaces, J. Math. Anal. Appl. 246 (2000), no. 1, 217-229. https://doi.org/10.1006/jmaa.2000.6783
  11. B. C. Grecu, Smooth 2-homogeneous polynomials on Hilbert spaces, Arch. Math. (Basel) 76 (2001), no. 6, 445-454. https://doi.org/10.1007/PL00000456
  12. B. C. Grecu, Geometry of 2-homogeneous polynomials on 𝑙p spaces, 1, J. Math. Anal. Appl. 273 (2002), no. 2, 262-282. https://doi.org/10.1016/S0022-247X(02)00217-2
  13. B. C. Grecu, Extreme 2-homogeneous polynomials on Hilbert spaces, Quaest. Math. 25 (2002), no. 4, 421-435. https://doi.org/10.2989/16073600209486027
  14. B. C. Grecu, Geometry of homogeneous polynomials on two-dimensional real Hilbert spaces, J. Math. Anal. Appl. 293 (2004), no. 2, 578-588. https://doi.org/10.1016/j.jmaa.2004.01.020
  15. B. C. Grecu, G. A. Munoz-Fernandez, and J. B. Seoane-Sepulveda, The unit ball of the complex 𝓟(3H), Math. Z. 263 (2009), no. 4, 775-785. https://doi.org/10.1007/s00209-008-0438-y
  16. R. B. Holmes, Geometric functional analysis and its applications, Graduate Texts in Mathematics, No. 24, Springer-Verlag, New York, 1975.
  17. S. G. Kim, Exposed 2-homogeneous polynomials on 𝓟(2𝑙2P) for 1 ≤ p ≤ ∞, Math. Proc. R. Ir. Acad. 107 (2007), no. 2, 123-129. https://doi.org/10.3318/PRIA.2007.107.2.123
  18. S. G. Kim, The unit ball of 𝓛s(2𝑙2), Extracta Math. 24 (2009), no. 1, 17-29.
  19. S. G. Kim, The unit ball of 𝓟(2D*(1, W)2), Math. Proc. R. Ir. Acad. 111A (2011), no. 2, 79-94. https://doi.org/10.3318/pria.2011.111.1.9
  20. S. G. Kim, The unit ball of 𝓛s(2d*(1, w)2), Kyungpook Math. J. 53 (2013), no. 2, 295-306. https://doi.org/10.5666/KMJ.2013.53.2.295
  21. S. G. Kim, Smooth polynomials of 𝓟(2D*(1, W)2), Math. Proc. R. Ir. Acad. 113A (2013), no. 1, 45-58. https://doi.org/10.3318/PRIA.2013.113.05
  22. 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
  23. S. G. Kim, Exposed symmetric bilinear forms of 𝓛s(2d*(1, w)2), Kyungpook Math. J. 54 (2014), no. 3, 341-347. https://doi.org/10.5666/KMJ.2014.54.3.341
  24. S. G. Kim, Polarization and unconditional constants of 𝓟(2d*(1, w)2), Commun. Korean Math. Soc. 29 (2014), no. 3, 421-428. https://doi.org/10.4134/CKMS.2014.29.3.421
  25. 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
  26. S. G. Kim, Exposed 2-homogeneous polynomials on the two-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
  27. S. G. Kim, The unit ball of 𝓛(22h(w)), Bull. Korean Math. Soc. 54 (2017), no. 2, 417-428. https://doi.org/10.4134/BKMS.b150851
  28. S. G. Kim, Extremal problems for 𝓛s(22h(w)), Kyungpook Math. J. 57 (2017), no. 2, 223-232. https://doi.org/10.5666/KMJ.2017.57.2.223
  29. S. G. Kim, The unit ball of 𝓛s(2𝑙3), Comment. Math. 57 (2017), no. 1, 1-7. https://doi.org/10.14708/cm.v57i1.1230
  30. S. G. Kim, The geometry of 𝓛s(3𝑙2), Commun. Korean Math. Soc. 32 (2017), no. 4, 991-997. https://doi.org/10.4134/CKMS.c170016
  31. S. G. Kim, Extreme 2-homogeneous polynomials on the plane with a hexagonal norm and applications to the polarization and unconditional constants, Studia Sci. Math. Hungar. 54 (2017), no. 3, 362-393. https://doi.org/10.1556/012.2017.54.3.1371
  32. S. G. Kim, The geometry of 𝓛(3𝑙2) and optimal constants in the Bohnenblust-Hille inequality for multilinear forms and polynomials, Extracta Math. 33 (2018), no. 1, 51-66. https://doi.org/10.17398/2605-5686.33.1.51
  33. S. G. Kim, Extreme bilinear forms on ℝn with the supremum norm, Period. Math. Hungar. 77 (2018), no. 2, 274-290. https://doi.org/10.1007/s10998-018-0246-z
  34. S. G. Kim, Exposed polynomials of ${\mathcal{P}}(^2{\mathbb{R}}^2_{h(\frac{1}{2})})$, Extracta Math. 33 (2018), no. 2, 127-143. https://doi.org/10.17398/2605-5686.33.2.127
  35. S. G. Kim, The unit ball of the space of bilinear forms on ℝ3 with the supremum norm, Commun. Korean Math. Soc. 34 (2019), no. 2, 487-494. https://doi.org/10.4134/CKMS.c180111
  36. S. G. Kim, Smooth points of 𝓛s(n𝑙2), Bull. Korean Math. Soc. 57 (2020), no. 2, 443-447. https://doi.org/10.4134/BKMS.b190311
  37. S. G. Kim, Extreme points of the space 𝓛(2𝑙), Commun. Korean Math. Soc. 35 (2020), no. 3, 799-807. https://doi.org/10.4134/CKMS.c190300
  38. S. G. Kim, Extreme points, exposed points and smooth points of the space 𝓛s(2𝑙3), Kyungpook Math. J. 60 (2020), no. 3, 485-505. https://doi.org/10.5666/KMJ.2020.60.3.485
  39. S. G. Kim, The unit balls of 𝓛(n𝑙m) and 𝓛s(n𝑙m), Studia Sci. Math. Hungar. 57 (2020), no. 3, 267-283. https://doi.org/10.1556/012.2020.57.3.1470
  40. S. G. Kim, Extreme and exposed points of 𝓛(n𝑙2) and 𝓛s(n𝑙2), Extracta Math. 35 (2020), no. 2, 127-135. https://doi.org/10.17398/2605-5686.35.2.127
  41. S. G. Kim, Extreme and exposed symmetric bilinear forms on the space 𝓛s(2𝑙2), Carpathian Math. Publ. 12 (2020), no. 2, 340-352. https://doi.org/10.15330/cmp.12.2.340-352
  42. S. G. Kim, Smooth points of 𝓛(n𝑙m) and 𝓛s(n𝑙m), Comment. Math. 60 (2020), no. 1-2, 13-21.
  43. S. G. Kim, Geometry of multilinear forms on ℝm with a certain norm, Acta Sci. Math. (Szeged) 87 (2021), no. 1-2, 233-245. https://doi.org/10.14232/actasm-020-824-2
  44. S. G. Kim, Geometry of multilinear forms on 𝑙1, Acta Comment. Univ. Tartu. Math. 25 (2021), no. 1, 57-66. https://doi.org/10.12697/acutm.2021.25.04
  45. 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
  46. A. G. Konheim and T. J. Rivlin, Extreme points of the unit ball in a space of real polynomials, Amer. Math. Monthly 73 (1966), 505-507. https://doi.org/10.2307/2315472
  47. M. Krein and D. Milman, On extreme points of regular convex sets, Studia Math. 9 (1940), 133-138. https://doi.org/10.4064/sm-9-1-133-138
  48. L. Milev and N. Naidenov, Strictly definite extreme points of the unit ball in a polynomial space, C. R. Acad. Bulgare Sci. 61 (2008), no. 11, 1393-1400.
  49. L. Milev and N. Naidenov, Semidefinite extreme points of the unit ball in a polynomial space, J. Math. Anal. Appl. 405 (2013), no. 2, 631-641. https://doi.org/10.1016/j.jmaa.2013.04.026
  50. G. A. Munoz-Fernandez, D. Pellegrino, J. B. Seoane-Sepulveda, and A. Weber, Supremum norms for 2-homogeneous polynomials on circle sectors, J. Convex Anal. 21 (2014), no. 3, 745-764.
  51. G. A. Munoz-Fernandez, S. Gy. Revesz, and J. B. Seoane-Sepulveda, Geometry of homogeneous polynomials on non symmetric convex bodies, Math. Scand. 105 (2009), no. 1, 147-160. https://doi.org/10.7146/math.scand.a-15111
  52. 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
  53. S. Neuwirth, The maximum modulus of a trigonometric trinomial, J. Anal. Math. 104 (2008), 371-396. https://doi.org/10.1007/s11854-008-0028-2
  54. 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