Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Extremal multilinear forms on Banach spaces

Author: I. Sarantopoulos
Journal: Proc. Amer. Math. Soc. 99 (1987), 340-346
MSC: Primary 46B20; Secondary 46G20
MathSciNet review: 870797
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that $L$ is a continuous symmetric $m$-linear form defined on a complex Banach space $E$, and $\hat L$ is the associated homogeneous polynomial. If \[ || L || = ({m^m}/m!)|| {\hat L} ||,\] we prove that $E$ contains an almost isometric copy of $l_m^1$. In particular if $E$ is an $m$-dimensional space, then $E$ is isometrically isomorphic to $l_m^1$. We also prove that the only examples of such extremal $L$ which achieve their norm are suitable "extensions" of a known example given by Nachbin.

References [Enhancements On Off] (What's this?)

  • Soo Bong Chae, Holomorphy and calculus in normed spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 92, Marcel Dekker, Inc., New York, 1985. With an appendix by Angus E. Taylor. MR 788158
  • Lawrence A. Harris, Bounds on the derivatives of holomorphic functions of vectors, Analyse fonctionnelle et applications (Comptes Rendus Colloq. Analyse, Inst. Mat., Univ. Federal Rio de Janeiro, Rio de Janeiro, 1972) Hermann, Paris, 1975, pp. 145–163. ActualitĂ©s Aci. Indust., No. 1367. MR 0477773
  • Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
  • R. S. Martin, Thesis, California Inst, of Tech., 1932.
  • I. Sarantopoulos, Estimates for polynomial norms on $L^p(\mu )$ spaces, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 2, 263–271. MR 817668, DOI
  • Seminaire Maurey-Schwartz, ExposĂ© VII, 1973-74.
  • Andrew Tonge, Polarization and the two-dimensional Grothendieck inequality, Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 2, 313–318. MR 735372, DOI
  • R. Daniel Mauldin (ed.), The Scottish Book, BirkhĂ€user, Boston, Mass., 1981. Mathematics from the Scottish CafĂ©; Including selected papers presented at the Scottish Book Conference held at North Texas State University, Denton, Tex., May 1979. MR 666400

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B20, 46G20

Retrieve articles in all journals with MSC: 46B20, 46G20

Additional Information

Article copyright: © Copyright 1987 American Mathematical Society