A Dvoretsky theorem for polynomials

Dineen, Seán

doi:10.1090/S0002-9939-1995-1264807-1

A Dvoretsky theorem for polynomials
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Proc. Amer. Math. Soc. 123 (1995), 2817-2821 Request permission

Abstract:

We lift upper and lower estimates from linear functionals to n-homogeneous polynomials and using this result show that ${l_\infty }$ is finitely represented in the space of n-homogeneous polynomials, $n \geq 2$, on any infinite-dimensional Banach space. Refinements are also given.

References

Raymundo Alencar, Richard M. Aron, and Seán Dineen, A reflexive space of holomorphic functions in infinitely many variables, Proc. Amer. Math. Soc. 90 (1984), no. 3, 407–411. MR 728358, DOI 10.1090/S0002-9939-1984-0728358-5
Richard M. Aron and Seán Dineen, $Q$-reflexive Banach spaces, Rocky Mountain J. Math. 27 (1997), no. 4, 1009–1025. MR 1627646, DOI 10.1216/rmjm/1181071856
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
Seán Dineen, Complex analysis in locally convex spaces, Notas de Matemática [Mathematical Notes], vol. 83, North-Holland Publishing Co., Amsterdam-New York, 1981. MR 640093
Aryeh Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 123–160. MR 0139079
Jeff D. Farmer, Polynomial reflexivity in Banach spaces, Israel J. Math. 87 (1994), no. 1-3, 257–273. MR 1286830, DOI 10.1007/BF02772998
Jeff Farmer and William B. Johnson, Polynomial Schur and polynomial Dunford-Pettis properties, Banach spaces (Mérida, 1992) Contemp. Math., vol. 144, Amer. Math. Soc., Providence, RI, 1993, pp. 95–105. MR 1209450, DOI 10.1090/conm/144/1209450
Manuel González and Joaquín M. Gutiérrez, Unconditionally converging polynomials on Banach spaces, Math. Proc. Cambridge Philos. Soc. 117 (1995), no. 2, 321–331. MR 1307084, DOI 10.1017/S030500410007314X

Compact polynomials between Banach spaces

J. A. Jaramillo and A. Prieto, Weak-polynomial convergence on a Banach space, Proc. Amer. Math. Soc. 118 (1993), no. 2, 463–468. MR 1126196, DOI 10.1090/S0002-9939-1993-1126196-0
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
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, DOI 10.1007/978-3-662-35347-9
Gilles Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. MR 1036275, DOI 10.1017/CBO9780511662454