## Extremal multilinear forms on Banach spaces

HTML articles powered by AMS MathViewer

- by I. Sarantopoulos PDF
- Proc. Amer. Math. Soc.
**99**(1987), 340-346 Request permission

## 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

- 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) Actualités Aci. Indust., No. 1367, Hermann, Paris, 1975, pp. 145–163. 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 10.1017/S0305004100064185
Seminaire Maurey-Schwartz, Exposé 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 10.1017/S0305004100061570 - R. Daniel Mauldin (ed.),
*The Scottish Book*, Birkhäuser, Boston, Mass., 1981. Mathematics from the Scottish Café; Including selected papers presented at the Scottish Book Conference held at North Texas State University, Denton, Tex., May 1979. MR**666400**

## Additional Information

- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**99**(1987), 340-346 - MSC: Primary 46B20; Secondary 46G20
- DOI: https://doi.org/10.1090/S0002-9939-1987-0870797-9
- MathSciNet review: 870797