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Locally Convex Spaces with Toeplitz Decompositions

Published online by Cambridge University Press:  09 April 2009

Juan M. Virués
Affiliation:
Escuela Superior de Ingenieros Camino de los Descubrimientos s/n 41092-SevillaSpain e-mail: [email protected], [email protected]
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Abstract

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A Toeplitz decomposition of a locally convez space E into subspaces (Ek) with continuous projections (Pk) is a decomposition of every x ∈ E as x = ΣkPkx where ordinary summability has been replaced by summability with respect to an infinite and row-finite matrix. We extend to the setting of Toeplitz decompositions a number of results about the locally convex structure of a space with a Schauder decomposition. Namely, we give some necessary or sufficient conditions for being reflexive, a Montel space or a Schwartz space. Roughly speaking, each of these locally convex properties is linked to a property of the convergence of the decomposition. We apply these results to study some structural questions in projective tensor products and spaces with Cesàro bases.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2000

References

[BMS]Bierstedt, K. D., Summers, W. H., ‘Köthe sets and Köthe sequence spaces’, in Functional analysis, holomorphy and approximation theory, Math. Studies, vol. 71 (Elsevier/North-Holland, Amsterdam, 1982) pp. 2791.Google Scholar
[BDe]Bonet, J. and Defant, A., ‘Projective tensor products of distingished Fréchet spaces’, Proc. Roy. Irish Acad. 85A (1985), 193199.Google Scholar
[BD]Bonet, J. and Díaz, J. C., ‘The problem of topologies of Grothendieck and the class of Fréchet T-spaces’, Math. Nachr. 150 (1991), 109118.CrossRefGoogle Scholar
[BDT]Bonet, J., Díaz, J. C. and Taskinen, J., ‘Tensor stable Fréchet and (DF)-spaces’, Collect. Math. 42 (1991), 199236.Google Scholar
[Bu1]Buntinas, M., ‘On sectionally dense summability fields’, Math. Z. 132 (1973), 141149.CrossRefGoogle Scholar
[Bu2]Buntinas, M., ‘On Toeplitz sections in sequence spaces’, Math. Proc. Cambridge Philos. Soc. 78 (1975), 451460.CrossRefGoogle Scholar
[DM1]Díaz, J. C. and Miñarro, M. A., ‘Distinguished Fréchet spaces and prohective tensor product’, Doga Mat. 14 (1990), 191208.Google Scholar
[DM2]Díaz, J. C. and Miñarro, M. A., ‘On Fréchet Montel spaces and their projective tensor produc’, Math. Proc. Cambridge Philos. Soc. 113 (1993), 335341.CrossRefGoogle Scholar
[F1]Florencio, M., ‘Sobre C-dualidad y espacios C-perfectos’, Rev. Real Acad. Cienc. Exact. Fís. Nature. Madrid 75 (1981), 12211234 (in Spanish).Google Scholar
[F2]Florencio, M., ‘Sobre la propiedad AK y AK-C en algunos espacios de sucesionesRev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 76 (1982), 575584 (in Spanish).Google Scholar
[F3]Florencio, M., ‘Una nota sobre especios escalonados generales’, Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 77 (1983), 221226 (in Spanish).Google Scholar
[FPe1]Florencio, M. and Carreras, P. Pérez, ‘Sobre sumabilidad Cesáro en el espacio CS (I)’, Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 75 (1981), 11851197 (in Spanish).Google Scholar
[FPe2]Florencio, M. and Carretas, P. Pérez, ‘Sobre sumabilidad Cesàro en el CS(II)’, Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 76 (1982), 531546 (in Spanish).Google Scholar
[FP]Florencio, M. and Paúl, P. J., ‘Some results on diagonal maps’, Bull. Soc. Roy. Sci. Liège 55 (1986), 569580.Google Scholar
[GW]Goes, S. and Welland, R., ‘Compactness criteria for Köthe spaces’, Math. Ann. 188 (1970), 251269.CrossRefGoogle Scholar
[Gr]Grothendieck, A., Produits tensoriels topologiques et espaces nucléaires Mem. Amer. Math. Soc. 16 (Amer. Math. Soc., Providence, 1955).Google Scholar
[Is]Isidro, J. M., ‘Quasinormability of some spaces of holomorphic mappings’, Rev. Mat. Univ. Complutense Madrid 3 (1990), 57123.Google Scholar
[Ja]James, R. C., ‘Bases and reflexivity of Banach spaces’, Ann. of Math. 52 (1950), 518527.CrossRefGoogle Scholar
[Jr]Jarchow, H., Locally convex spaces (B. G. Teubner, Stuttgart, 1981).CrossRefGoogle Scholar
[Ka]Kalton, N. J., ‘Schauder decompositions of locally convex spaces’, Math. Proc. Cambridge Philos. Soc. 68 (1970), 377392.CrossRefGoogle Scholar
[Ke]Kelley, J. L., General topology (Springer, Berlin, 1975).Google Scholar
[Kö]Köthe, G., Topological vector spaces I, II (Springer, Berlin, 1969, 1979).Google Scholar
[Me]Mayers, G., ‘On Toeplitz sections in FK-spaces’, Studia Math. 51 (1974), 2333.CrossRefGoogle Scholar
[Mi]Miñarro, M. A., Descomposiciones de espacios de Fréchet. Aplicación al producto tensorial proyectivo (Ph.D. Thesis, Universidad de Sevilla, Seville, 1991), (in Spanish).Google Scholar
[PSV1]Paúl, P. J., Sáez, C. and Virués, J. M., ‘Completeness of spaces with Toeplitz decompositions’, Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 92 (1998), 2733.Google Scholar
[PSV2]Paúl, P. J., Sáez, C. and Virués, J. M., ‘Barrelledness of spaces with Toeplitz decompositions’, preprint, 1998.CrossRefGoogle Scholar
[PeB]Carreras, P. Pérez and Bonet, J., Barrelled locally convex spaces, Mathematics Studies, vol. 131 (North-Holland, Amsterdam, 1987).CrossRefGoogle Scholar
[Pr]Peris, A., ‘Quasinormable spaces and the problem of topologies of Grothendieck’, Ann. Acad. Sci. Fenn. Ser. A I. Math. 19 (1994), 167203.Google Scholar
[Ru]Ruckle, W. H., Sequence spaces, Research Notes in Math., vol. 49 (Pitman, London, 1981).Google Scholar
[Si1]Singer, I., ‘Basic sequences and reflexivity of Banach spaces’, Studia Math. 21 (1961/1962), 351369.CrossRefGoogle Scholar
[Si2]Singer, I., Bases in Banach spaces I, II (Springer, Berlin, 1970, 1981).CrossRefGoogle Scholar
[T]Taskinen, J., ‘The projective tensor products of Fréchet-Montel spaces’, Studia Math. 91 (1988), 1730.CrossRefGoogle Scholar
[V]Valdivia, M., Topics in locally convex spaces, vol. 67 (North-Holland, Amsterdam, 1982).CrossRefGoogle Scholar
[We]Webb, J. H., ‘Schauder basis and decompositions in locally convex spaces’, Math. Proc. Cambridge Philos. Soc. 76 (1974), 145152.CrossRefGoogle Scholar
[Wi1]Wilansky, A., Modern methods in topological vector spaces (McGraw-Hill, New York, 1978).Google Scholar
[Wi2]Wilansky, A., Summability through functional analysis, Math. Studies, vol. 85 (North-Holland, Amsterdam, 1984).Google Scholar
[Ze]Zeller, K., ‘Approximation in wirkfeldern von Summierungsverfahren”, Arch. Math. (Basel) 4 (1953), 425431.CrossRefGoogle Scholar