Non-Archimedean John, Non-Archimedean Dvoretzky-Milman, Non-Archimedean Type-Cotype and Non-Archimedean Kwapien Problems

Primarily human-written textHuman understanding: all partsmath.NT — Number Theorymath.FA — Functional Analysis

Contributed by K. MAHESH KRISHNA ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We ask for non-Archimedean version of following four: (1) John Theorem, (2) Dvoretzky-Milman Theorem, (3) Type-Cotype of Banach space, (4) Kwapien Theorem.

References

  1. Fernando Albiac and Nigel J. Kalton. Topics in B anach space theory , volume 233 of Graduate Texts in Mathematics . Springer, 2016.
  2. Paolo Aniello, Stefano Mancini, and Vincenzo Parisi. Trace class operators and states in pp -adic quantum mechanics. J. Math. Phys. , 64(5):45, 2023.
  3. Paolo Aniello, Stefano Mancini, and Vincenzo Parisi. Quantum mechanics on a pp -adic Hilbert space: foundations and prospects. Int. J. Geom. Methods Mod. Phys. , 21(10):36, 2024.
  4. Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman. Asymptotic geometric analysis. I , volume 202 of Math. Surv. Monogr. Providence, RI: American Mathematical Society (AMS), 2015.
  5. Joe Diestel, Hans Jarchow, and Andrew Tonge. Absolutely summing operators , volume 43 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 1995.
  6. Aryeh Dvoretzky. Some results on convex bodies and B anach spaces. In Proc. I nternat. S ympos. L inear S paces ( J erusalem, 1960) , pages 123--160. Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961.
  7. T. Figiel. Some remarks on D voretzky's theorem on almost spherical sections of convex bodies. Colloq. Math. , 24:241--252, 1971/72.
  8. T. Figiel. A short proof of D voretzky's theorem. In Séminaire M aurey- S chwartz 1974--1975: E spaces LpL^{p} , applications radonifiantes et géométrie des espaces de B anach, E xp. N o. XXIII . 1975.
  9. T. Figiel. A short proof of D voretzky's theorem on almost spherical sections of convex bodies. Compositio Math. , 33(3):297--301, 1976.
  10. T. Figiel, J. Lindenstrauss, and V. D. Milman. The dimension of almost spherical sections of convex bodies. Acta Math. , 139(1-2):53--94, 1977.
Show all 37 referencesShow fewer references
  1. Y. Gordon, O. Guedon, and M. Meyer. An isomorphic D voretzky's theorem for convex bodies. Studia Math. , 127(2):191--200, 1998.
  2. Yehoram Gordon. Some inequalities for Gaussian processes and applications. Isr. J. Math. , 50:265--289, 1985.
  3. Yehoram Gordon. Gaussian processes and almost spherical sections of convex bodies. Ann. Probab. , 16(1):180--188, 1988.
  4. Fritz John. Extremum problems with inequalities as subsidiary conditions. Studies Essays , pres. to R . Courant , 187-204 (1948)., 1948.
  5. W. B. Johnson and J. Lindenstrauss, editors. Handbook of the geometry of B anach spaces. V ol. I . North-Holland Publishing Co., Amsterdam, 2001.
  6. W. B. Johnson and J. Lindenstrauss, editors. Handbook of the geometry of B anach spaces. V ol. 2 . North-Holland, Amsterdam, 2003.
  7. P. Jordan and J. von Neumann. On inner products in linear, metric spaces. Ann. Math. (2) , 36:719--723, 1935.
  8. G. K. Kalisch. On p-adic Hilbert spaces. Ann. Math. (2) , 48:180--192, 1947.
  9. S. Kwapie ń . Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. Stud. Math. , 44:583--595, 1972.
  10. Rafal Latala and Krzysztof Oleszkiewicz. On the best constant in the K hinchin- K ahane inequality. Studia Math. , 109(1):101--104, 1994.
  11. Daniel Li and Herve Queffelec. Introduction to B anach spaces: analysis and probability. V ol. 1 , volume 166 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2018.
  12. Daniel Li and Herve Queffelec. Introduction to B anach spaces: analysis and probability. V ol. 2 , volume 167 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2018.
  13. J. Matousek. Lectures on discrete geometry , volume 212 of Graduate Texts in Mathematics . Springer-Verlag, New York, 2002.
  14. Ivan Matsak and Anatolij Plichko. Dvoretzky's theorem by G aussian method. In Functional analysis and its applications , volume 197 of North-Holland Math. Stud. , pages 171--184. Elsevier Sci. B. V., Amsterdam, 2004.
  15. Bernard Maurey. Type, cotype and KK -convexity. In Handbook of the geometry of Banach spaces. Volume 2 , pages 1299--1332. Amsterdam: North-Holland, 2003.
  16. V. Milman. Dvoretzky's theorem---thirty years later. Geom. Funct. Anal. , 2(4):455--479, 1992.
  17. V. D. Milman. A new proof of A . D voretzky's theorem on cross-sections of convex bodies. Funkcional. Anal. i Priložen. , 5(4):28--37, 1971.
  18. Vitali D. Milman and Gideon Schechtman. Asymptotic theory of finite-dimensional normed spaces , volume 1200 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1986.
  19. C. Perez-Garcia and W. H. Schikhof. Locally convex spaces over non- Archimedean valued fields , volume 119 of Camb. Stud. Adv. Math. Cambridge: Cambridge University Press, 2010.
  20. Gilles Pisier. Factorization of linear operators and geometry of Banach spaces , volume 60 of Reg. Conf. Ser. Math. Providence, RI: American Mathematical Society (AMS), 1986.
  21. Gilles Pisier. Probabilistic methods in the geometry of B anach spaces. In Probability and analysis ( V arenna, 1985) , volume 1206 of Lecture Notes in Math. , pages 167--241. Springer, Berlin, 1986.
  22. Gilles Pisier. The volume of convex bodies and B anach space geometry , volume 94 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 1989.
  23. Gideon Schechtman. A remark concerning the dependence on ϵ\epsilon in D voretzky's theorem. In Geometric aspects of functional analysis (1987--88) , volume 1376 of Lecture Notes in Math. , pages 274--277. Springer, Berlin, 1989.
  24. W. H. Schikhof. Ultrametric calculus. An introduction to pp -adic analysis , volume 4 of Camb. Stud. Adv. Math. Cambridge: Cambridge University Press, 2006.
  25. Andrzej Szankowski. On D voretzky's theorem on almost spherical sections of convex bodies. Israel J. Math. , 17:325--338, 1974.
  26. S. J. Szarek. On the best constants in the K hinchin inequality. Studia Math. , 58(2):197--208, 1976.
  27. Yasuo Yamasaki. A simple proof of Kwapien 's theorem. Publ. Res. Inst. Math. Sci. , 20:1247--1251, 1984.

Version history

  1. v1Submitted by K. MAHESH KRISHNAInitial depositCurrentOct 29, 2026