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Banach spaces of homogeneous polynomials without the approximation property

Seán Dineen; Jorge Mujica

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 2, page 367-374
  • ISSN: 0011-4642

Abstract

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We present simple proofs that spaces of homogeneous polynomials on L p [ 0 , 1 ] and ℓ p provide plenty of natural examples of Banach spaces without the approximation property. By giving necessary and sufficient conditions, our results bring to completion, at least for an important collection of Banach spaces, a circle of results begun in 1976 by R. Aron and M. Schottenloher (1976).

How to cite

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Dineen, Seán, and Mujica, Jorge. "Banach spaces of homogeneous polynomials without the approximation property." Czechoslovak Mathematical Journal 65.2 (2015): 367-374. <http://eudml.org/doc/270087>.

@article{Dineen2015,
abstract = {We present simple proofs that spaces of homogeneous polynomials on $L_\{p\}[0,1]$ and $\ell _\{p\}$ provide plenty of natural examples of Banach spaces without the approximation property. By giving necessary and sufficient conditions, our results bring to completion, at least for an important collection of Banach spaces, a circle of results begun in 1976 by R. Aron and M. Schottenloher (1976).},
author = {Dineen, Seán, Mujica, Jorge},
journal = {Czechoslovak Mathematical Journal},
keywords = {Banach space; approximation property; linear operator; homogeneous polynomial; holomorphic function; Banach spaces; approximation property; linear operators; homogeneous polynomials; holomorphic functions},
language = {eng},
number = {2},
pages = {367-374},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Banach spaces of homogeneous polynomials without the approximation property},
url = {http://eudml.org/doc/270087},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Dineen, Seán
AU - Mujica, Jorge
TI - Banach spaces of homogeneous polynomials without the approximation property
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 2
SP - 367
EP - 374
AB - We present simple proofs that spaces of homogeneous polynomials on $L_{p}[0,1]$ and $\ell _{p}$ provide plenty of natural examples of Banach spaces without the approximation property. By giving necessary and sufficient conditions, our results bring to completion, at least for an important collection of Banach spaces, a circle of results begun in 1976 by R. Aron and M. Schottenloher (1976).
LA - eng
KW - Banach space; approximation property; linear operator; homogeneous polynomial; holomorphic function; Banach spaces; approximation property; linear operators; homogeneous polynomials; holomorphic functions
UR - http://eudml.org/doc/270087
ER -

References

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  1. Alencar, R., On reflexivity and basis for P ( m E ) , Proc. R. Ir. Acad., Sect. A 85 (1985), 131-138. (1985) MR0845536
  2. Arias, A., Farmer, J. D., 10.2140/pjm.1996.175.13, Pac. J. Math. 175 (1996), 13-37. (1996) MR1419470DOI10.2140/pjm.1996.175.13
  3. Aron, R. M., Schottenloher, M., 10.1016/0022-1236(76)90026-4, J. Funct. Anal. 21 (1976), 7-30. (1976) Zbl0328.46046MR0402504DOI10.1016/0022-1236(76)90026-4
  4. Banach, S., Théorie des Opérations Linéaires, Chelsea Publishing Co. New York French (1955). (1955) Zbl0067.08902MR0071726
  5. Coeuré, G., 10.5802/aif.345, Ann. Inst. Fourier 20 French (1970), 361-432. (1970) Zbl0187.39003MR0274804DOI10.5802/aif.345
  6. Defant, A., Floret, K., Tensor Norms and Operator Ideals, North-Holland Mathematics Studies 176 North-Holland, Amsterdam (1993). (1993) Zbl0774.46018MR1209438
  7. Díaz, J. C., Dineen, S., 10.1007/BF02385668, Ark. Mat. 36 (1998), 87-96. (1998) Zbl0929.46036MR1611149DOI10.1007/BF02385668
  8. Diestel, J., Jarchow, H., Tonge, A., Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics 43 Cambridge Univ. Press, Cambridge (1995). (1995) Zbl0855.47016MR1342297
  9. J. Diestel, J. J. Uhl, Jr., Vector Measures, Mathematical Surveys 15 American Mathematical Society, Providence (1977). (1977) Zbl0369.46039MR0453964
  10. Dineen, S., 10.1007/978-1-4471-0869-6, Springer Monographs in Mathematics Springer, London (1999). (1999) Zbl1034.46504MR1705327DOI10.1007/978-1-4471-0869-6
  11. Dineen, S., Mujica, J., 10.1007/s13398-012-0065-7, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 106 (2012), 457-469. (2012) Zbl1266.46036MR2978926DOI10.1007/s13398-012-0065-7
  12. Dineen, S., Mujica, J., 10.1016/j.jfa.2010.04.001, J. Funct. Anal. 259 (2010), 545-560. (2010) Zbl1195.46038MR2644113DOI10.1016/j.jfa.2010.04.001
  13. Dineen, S., Mujica, J., 10.1016/j.jat.2004.01.008, J. Approx. Theory 126 (2004), 141-156. (2004) Zbl1060.41031MR2045536DOI10.1016/j.jat.2004.01.008
  14. Enflo, P., 10.1007/BF02392270, Acta Math. 130 (1973), 309-317. (1973) Zbl0267.46012MR0402468DOI10.1007/BF02392270
  15. Floret, K., Natural norms on symmetric tensor products of normed spaces. Proceedings of the Second International Workshop on Functional Analysis, Trier, 1997, Note Mat. 17 (1997), 153-188. (1997) MR1749787
  16. Gelbaum, B. R., Lamadrid, J. G. de, 10.2140/pjm.1961.11.1281, Pac. J. Math. 11 (1961), 1281-1286. (1961) Zbl0106.08604MR0147881DOI10.2140/pjm.1961.11.1281
  17. Godefroy, G., Saphar, P. D., 10.1090/S0002-9939-1989-0930249-6, Proc. Am. Math. Soc. 105 (1989), 70-75. (1989) Zbl0674.46009MR0930249DOI10.1090/S0002-9939-1989-0930249-6
  18. Grothendieck, A., Produits Tensoriels Topologiques et Espaces Nucléaires, Mem. Am. Math. Soc. 16 French (1955), 140 pages. (1955) Zbl0123.30301MR0075539
  19. Mujica, J., Complex Analysis in Banach Spaces. Holomorphic Functions and Domains of Holomorphy in Finite and Infinite Dimensions, North-Holland Math. Stud. 120. Notas de Matemática 107 North-Holland, Amsterdam (1986). (1986) Zbl0586.46040MR0842435
  20. Mujica, J., Spaces of holomorphic functions and the approximation property, Lecture Notes, Universidad Complutense de Madrid, 2009. 
  21. Nachbin, L., Sur les espaces vectoriels topologiques d'applications continues, C. R. Acad. Sci., Paris, Sér. A 271 French (1970), 596-598. (1970) Zbl0205.12402MR0271712
  22. Nachbin, L., On the topology of the space of all holomorphic functions on a given open subset, Nederl. Akad. Wet., Proc., Ser. A 70 Indag. Math. 29 (1967), 366-368. (1967) Zbl0147.11402MR0215066
  23. Pełczyński, A., 10.4064/sm-19-2-209-228, Stud. Math. 19 (1960), 209-228. (1960) Zbl0104.08503MR0126145DOI10.4064/sm-19-2-209-228
  24. Pełczyński, A., 10.4064/sm-16-2-173-182, Stud. Math. 16 (1957), 173-182. (1957) Zbl0080.09701MR0093698DOI10.4064/sm-16-2-173-182
  25. Pietsch, A., History of Banach Spaces and Linear Operators, Birkhäuser Basel (2007). (2007) Zbl1121.46002MR2300779
  26. Pisier, G., De nouveaux espaces de Banach sans la propriété d'approximation (d'après A. Szankowski), Séminaire Bourbaki 1978/79 Lecture Notes in Math. 770 Springer, Berlin French (1980), 312-327. (1980) Zbl0443.46015MR0572431
  27. Szankowski, A., 10.1007/BF02392870, Acta Math. 147 (1981), 89-108. (1981) MR0631090DOI10.1007/BF02392870

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