Polynomial continuity on

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

DOI:
https://doi.org/10.1090/S0002-9939-97-03733-7

MathSciNet review:
1371124

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Abstract | References | Similar Articles | Additional Information

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 has property (RP) if given two bounded sequences , we have that for every polynomial on whenever for every polynomial on ; i.e., the restriction of every polynomial on 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 must be polynomially continuous.

**1.**R. M. Aron, Y. S. Choi and J. G. Llavona, Estimates by polynomials,*Bull. Austral. Math. Soc.***52**(1995), 475-486. CMP**96:03****2.**R. M. Aron and J. B. Prolla, Polynomial approximation of differentiable functions on Banach spaces,*J. Reine Angew. Math.***313**(1980), 195-216. MR**81c:41078****3.**T. K. Carne, B. Cole and T. W. Gamelin, A uniform algebra of analytic functions on a Banach space,*Trans. Amer. Math. Soc.***314**(1989), 639-659. MR**90i:46098****4.**A. M. Davie and T. W. Gamelin, A theorem on polynomial-star approximation,*Proc. Amer. Math. Soc.***106**(1989), 351-356. MR**89k:46023****5.**J. Diestel,*Sequences and Series in Banach Spaces,*Graduate Texts in Math.**92**, Springer, Berlin 1984. MR**85i:46020****6.**J. Mujica,*Complex Analysis in Banach Spaces,*Math. Studies**120**, North-Holland, Amsterdam 1986. MR**88d:46084**

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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