Abstract
Countable projective limits of countable inductive limits, so-called PLB-spaces, of weighted Banach spaces of continuous functions have recently been investigated by Agethen et al., who analyzed locally convex properties in terms of the defining double sequence of weights. We complement their results by considering a defining sequence which is the product of two single sequences. By associating these two sequences with a weighted Fréchet, resp. LB-space of continuous functions or with two weighted Fréchet spaces (by taking the reciprocal of one of the sequences) we derive a representation of the PLB-space as the tensor product of a Fréchet and a DF-space and exhibit a connection between the invariants (DN) and (Ω) for Fréchet spaces and locally convex properties of the PLB-space resp. of the forementioned tensor product.
Similar content being viewed by others
References
Agethen, S.: Spaces of continuous and holomorphic functions with growth conditions, Dissertation Universität Paderborn (2004)
Agethen S., Bierstedt K.D., Bonet J.: Projective limits of weighted (LB)-spaces of continuous functions. Arch. Math. (Basel) 92(5), 384–398 (2009)
Bierstedt K.D.: Gewichtete Räume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt. I. J. Reine Angew. Math. 259, 186–210 (1973)
Bierstedt K.D.: Injektive Tensorprodukte und Slice-Produkte gewichteter Räume stetiger Funktionen. J. Reine Angew. Math. 266, 121–131 (1974)
Bierstedt, K.D.: An introduction to locally convex inductive limits, functional analysis and its applications (Nice, 1986) 35–133. In: ICPAM Lecture Notes, pp. 35–133. World Scientific Publishing, Singapore (1988)
Bierstedt K.D., Bonet J.: Weighted (LF)-spaces of continuous functions. Math. Nachr. 165, 25–48 (1994)
Bierstedt, K.D., Meise, R., Summers, W.H.: Köthe sets and Köthe sequence spaces. In: Functional analysis, holomorphy and approximation theory (Rio de Janeiro, 1980), vol. 71, pp. 27–91. North-Holland Mathematics Studies, Amsterdam (1982)
Bierstedt K.D., Meise R., Summers W.H.: A projective description of weighted inductive limits. Trans. Am. Math. Soc. 272(1), 107–160 (1982)
Bonet J., Domański P.: Parameter dependence of solutions of partial differential equations in spaces of real analytic functions. Proc. Am. Math. Soc. 129(2), 495–503 (2001)
Bonet J., Domański P.: Parameter dependence of solutions of differential equations on spaces of distributions and the splitting of short exact sequences. J. Funct. Anal. 230(2), 329–381 (2006)
Bonet J., Domański P.: The stucture of spaces of quasianalytic functions of Romieu type. Arch. Math. (Basel) 89(5), 430–441 (2007)
Bonet J., Domański P.: The splitting of exact sequences of PLS-spaces and smooth dependence of solutions of linear partial differential equations. Adv. Math. 217(2), 561–585 (2008)
Braun R., Vogt D.: A sufficient condition for Proj\({\,^1\fancyscript{X}=0}\) . Michigan Math. J. 44(1), 149–156 (1997)
Domański, P.: Classical PLS-spaces: spaces of distributions, real analytic functions and their relatives, Orlicz centenary volume, vol. 64. pp. 51–70. Polish Academy of Sciences, Warsaw (2004)
Domański P.: Real analytic parameter dependence of solutions of differential equations. Rev. Mat. Iberoamericana 26, 174–238 (2010)
Frerick L., Wengenroth J.: A sufficient condition for vanishing of the derived projective limit functor. Arch. Math. (Basel) 67(4), 296–301 (1996)
Grothendieck, A.: Produits Tensoriels Topologiques et Espaces Nuclé aires, Mem. Am. Math. Soc. 16 (1955)
Hollstein R.: Inductive limits and \({\varepsilon}\) -tensor products. J. Reine Angew. Math. 319, 38–62 (1980)
Jarchow H.: Locally Convex Spaces. B. G. Teubner, Stuttgart (1981)
Köthe G.: Topological vector spaces. I, Grundlehren der math. Wiss, vol. 159. Springer, New York (1969)
Köthe G.: Topological vector spaces. II, Grundlehren der math. Wiss, vol. 237. Springer, New York (1979)
Meise R., Vogt D.: Introduction to functional analysis. Oxford Graduate Texts in Math, vol. 2. The Clarendon Press/Oxford University Press, New York (1997)
Palamodov, V.P.: The projective limit functor in the category of topological linear spaces, Mat. Sb. 75, 567–603 (in Russian), English transl. Math. USSR Sbornik 17, 189–315 (1968)
Palamodov, V.P.: Homological methods in the theory of locally convex spaces, Uspekhi Mat. Nauk 26(1), 3–66 (in Russian), English transl., Russian Math. Surveys 26(1), 1–64 (1971)
Pérez Carreras P., Bonet J.: Barrelled locally convex spaces. vol. 113. North-Holland Mathematics Studies, Amsterdam (1987)
Piszczek K.: On a property of PLS-spaces inherited by their tensor products. Bull. Belg. Math. Soc. Simon Stevin 17, 155–170 (2010)
Retakh V.S.: Subspaces of a countable inductive limit (English transl.). Soviet Math. Dokl. 11, 1384–1386 (1971)
Varol O.: A generalization of a theorem of A. Grothendieck. Math. Nachr. 280(3), 313–325 (2007)
Vogt D.: Charakterisierung der Unterräume von s. Math. Z. 155(2), 109–117 (1977)
Vogt D.: Frécheträume zwischen denen jede stetige lineare Abbildung beschränkt ist. J. Reine Angew. Math. 345, 182–200 (1983)
Vogt, D.: Lectures on projective spectra of (DF)-spaces. In: Seminar lectures, Wuppertal (1987)
Vogt D.: On the functors Ext1(E,F) for Fréchet spaces. Studia Math. 85(2), 163–197 (1987)
Vogt, D.: Topics on projective spectra of (LB)-spaces, Adv. in the Theory of Fréchet spaces (Istanbul, 1988), NATO Adv. Sci. Inst. Ser. C 287, 11–27 (1989)
Vogt, D.: Regularity properties of (LF)-spaces, progress in functional analysis. In: Proc. Int. Meet. Occas. 60th Birthd. M. Valdivia, Peñscola/Spain, vol. 170, pp. 57–84. North-Holland Mathematics Studies (1992)
Vogt D., Wagner M.-J.: Charakterisierung der Quotientenräume von s und eine Vermutung von Martineau. Studia Math. 67(3), 225–240 (1980)
Wegner, S.-A.: Inductive kernels and projective hulls for weighted (PLB)- and (LF)-spaces of continuous functions, Diplomarbeit Universität Paderborn (2007)
Wegner, S.-A.: Projective limits of weighted (LB)-spaces of holomorphic functions, PhD-thesis, Universidad Politécnica de Valencia (2010)
Wengenroth, J.: Derived functors in functional analysis. In: Lecture Notes in Mathematics, vol. 1810. Springer, Berlin (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wegner, SA. Weighted PLB-spaces of continuous functions arising as tensor products of a Fréchet and a DF-space. RACSAM 105, 85–96 (2011). https://doi.org/10.1007/s13398-011-0001-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-011-0001-2