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Polynomial continuity on $\ell _1$


Authors: Manuel González, Joaquín M. Gutiérrez and José G. Llavona
Journal: Proc. Amer. Math. Soc. 125 (1997), 1349-1353
MSC (1991): Primary 46E15; Secondary 46B20
MathSciNet review: 1371124
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Abstract: A mapping between Banach spaces is said to be polynomially continuous if its restriction to any bounded set is uniformly continuous for the weak polynomial topology. A Banach space $X$ has property (RP) if given two bounded sequences $(u_j), (v_j)\subset X$, we have that $Q(u_j)-Q(v_j)\rightarrow 0$ for every polynomial $Q$ on $X$ whenever $P(u_j-v_j)\rightarrow 0$ for every polynomial $P$ on $X$; i.e., the restriction of every polynomial on $X$ to each bounded set is uniformly sequentially continuous for the weak polynomial topology. We show that property (RP) does not imply that every scalar valued polynomial on $X$ must be polynomially continuous.


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Additional Information

Manuel González
Email: gonzalem@ccaix3.unican.es

Joaquín M. Gutiérrez
Affiliation: Departamento de Matemáticas, ETS de Ingenieros Industriales, Universidad Politéc- nica de Madrid, C. José Gutiérrez Abascal 2, 28006 Madrid, Spain
Email: c0550003@ccupm.upm.es

José G. Llavona
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: llavona@eucmax.sim.ucm.es

DOI: https://doi.org/10.1090/S0002-9939-97-03733-7
Keywords: Polynomials on Banach spaces, weak polynomial topology, polynomials on $\ell_1$
Received by editor(s): October 30, 1995
Additional Notes: The first author was supported in part by DGICYT Project PB 94–1052 (Spain), and the second and third authors by DGICYT Project PB 93–0452 (Spain)
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society