Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-28T16:36:48.715Z Has data issue: false hasContentIssue false
  • English
  • Français

Locally Convex Spaces with Toeplitz Decompositions

Part of: Topological linear spaces and related structures

Published online by Cambridge University Press:  09 April 2009

Pedro J. Paúl ,
Carmen Sáez  and
Juan M. Virués
Juan M. Virués
Affiliation:
Escuela Superior de Ingenieros Camino de los Descubrimientos s/n 41092-SevillaSpain e-mail: piti@cica.es, virues@matinc.us.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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(Ek) with continuous projections (Pk) is a decomposition of every x ∈ E as x = ΣkPkx where ordinary summability has been replaced by summability with respect to an infinite and row-finite matrix.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.

Keywords

Decompositions of locally convex spacesreflexivemontel and Schwartz spacessummability and basestensor productssequence spaces

MSC classification

Secondary: 46A11: Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) 46A25: Reflexivity and semi-reflexivity 46A35: Summability and bases 46A45: Sequence spaces (including Köthe sequence spaces)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 1 , February 2000 , pp. 19 - 40
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. 27–91.Google Scholar
[BDe]Bonet, J. and Defant, A., ‘Projective tensor products of distingished Fréchet spaces’, Proc. Roy. Irish Acad. 85A (1985), 193–199.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), 109–118.CrossRefGoogle Scholar
[BDT]Bonet, J., Díaz, J. C. and Taskinen, J., ‘Tensor stable Fréchet and (DF)-spaces’, Collect. Math. 42 (1991), 199–236.Google Scholar
[Bu1]Buntinas, M., ‘On sectionally dense summability fields’, Math. Z. 132 (1973), 141–149.CrossRefGoogle Scholar
[Bu2]Buntinas, M., ‘On Toeplitz sections in sequence spaces’, Math. Proc. Cambridge Philos. Soc. 78 (1975), 451–460.CrossRefGoogle Scholar
[DM1]Díaz, J. C. and Miñarro, M. A., ‘Distinguished Fréchet spaces and prohective tensor product’, Doga Mat. 14 (1990), 191–208.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), 335–341.CrossRefGoogle Scholar
[F1]Florencio, M., ‘Sobre C-dualidad y espacios C-perfectos’, Rev. Real Acad. Cienc. Exact. Fís. Nature. Madrid 75 (1981), 1221–1234 (in Spanish).Google Scholar
[F2]Florencio, M., ‘Sobre la propiedad AK y AK-C en algunos espacios de sucesiones’ Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 76 (1982), 575–584 (in Spanish).Google Scholar
[F3]Florencio, M., ‘Una nota sobre especios escalonados generales’, Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 77 (1983), 221–226 (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), 1185–1197 (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), 531–546 (in Spanish).Google Scholar
[FP]Florencio, M. and Paúl, P. J., ‘Some results on diagonal maps’, Bull. Soc. Roy. Sci. Liège 55 (1986), 569–580.Google Scholar
[GW]Goes, S. and Welland, R., ‘Compactness criteria for Köthe spaces’, Math. Ann. 188 (1970), 251–269.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), 57–123.Google Scholar
[Ja]James, R. C., ‘Bases and reflexivity of Banach spaces’, Ann. of Math. 52 (1950), 518–527.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), 377–392.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), 23–33.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), 27–33.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), 167–203.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), 351–369.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), 17–30.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), 145–152.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), 425–431.CrossRefGoogle Scholar