Abstract
We consider locally convex inductive and projective limits of weighted normed Köthe function spaces of.measurable functions on a σ-finite measure space (X, Σ,μ). This general framework includes inductive and projective limits of weighted Lp-spaces, 1 ≤ p ≤ ∞, Lorentz spaces L p,q 1 < p < ∞, 1 ≤ q ≤ ∞, and Orlicz function spaces. In the projective case we characterize the quasinormability of L p(A) in terms of the increasing sequence A = (a n)n∈ℕ of weights on X. For the most important normed Köthe function spaces Lp, like Lp-spaces or L p,q-spaces, this condition on A = (an)n∈ℕ (V = (v n = 1/an)n∈ℕ) allows a useful projective description of the inductive limit topology of k p(V) by an associated system ofweights. A characterization of the Montel spaces L P(A) in terms of A = (an)n∈ℕ is also given. As an application we obtain functional analytic results on weighted inductive limits of spaces of holomorphic and harmonic functions.
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Reference
A. Baernstein II, Representation of holomorphic functions by boundary integrals, Trans. Amer. Math. Soc. 160 (1971), 27–37.
K.- D. Bierstedt, J. Bonet, Stefan Heinrich’s density condition for Fréchet spaces and the characterization of the distinguished Köthe echelon spaces, to appear in Math. Nachr. (also see: C. R. Acad. Sci. Paris 303, Série I, no. 10 (1986), 459–462).
K.- D. Bierstedt, R. Meise, Induktive Limites gewichteter Räume stetiger und holomorpher Funktionen, J. reine angew. Math. 282 (1976), 186–220.
K.- D. Bierstedt, R. Meise, Distinguished echelon spaces and the projective description of weighted inductive limits of type V dC(X), Aspects of Mathematics and its Applications, North-Holland Math. Library 34 (1986), 169–226.
K.- D. Bierstedt, R. Meise, W. H. Summers, A projective description of weighted inductive limits, Trans. Amer. Math. Soc. 272 (1982), 107–160.
K.- D. Bierstedt, R. G. Meise, W. H. Summers, Köthe sets and Köthe sequence spaces, Functional analysis, Holomorphy and Approximation Theory, North-Holland Math. Stud. 71 (1982), 27–91.
E. Dubinsky, Perfect Fréchet Spaces, Math. Ann. 174 (1967), 186–194.
A. Grothendieck, Sur les espaces (F) et (DF), Summa Brasil. Math. 3 (1954), 57–122.
S. Heinrich, Ultrapowers of locally convex spaces and applications I, Math. Nachr. 118 (1984), 285–315.
J. Horváth, Topological Vector Spaces and Distributions I, Addison-Wesley Series in Mathematics, Reading, Mass. (1966).
L. Hörmander, An Introduction To Complex Analysis In Several Variables, North-Holland Math. Library 7 (1973).
H. Jarchow, Locally Convex Spaces, B. G. Teubner (1981).
G. Köthe, Topological Vector Spaces I, Springer Grundlehren der math. Wiss. 159 (1969).
J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces II, Springer Ergebnisse der Math, und ihrer Grenzgebiete 97 (1979).
J. A. López Molina, Reflexividad en los espacios escalonados de Köthe, Rev. de la Real Academia de Ciencias Madrid 75 (1981), 213–232.
W. A. J. Luxemburg, A. C. Zaanen, Riesz Spaces I, North-Holland, Amsterdam-London (1971).
R. Meise, Halbnormensysteme für gewisse (LB)-Räume, unpublished manuscript.
A. Pietsch, Nuclear locally convex spaces (3rd ed.), Springer Ergebnisse der Math, und ihrer Grenzgebiete, vol. 66 (1972).
K. Reiher, Zur Theorie verallgemeinerter Köthescher Folgen- und Funktionenräume, Dissertation, Paderborn (1986).
K. Reiher, Zur Theorie verallgemeinerter Köthescher Folgenräume, preprint.
M. Valdivia, Topics in Locally Convex Spaces, North-Holland Math. Studies 67 (1982).
J. Wloka, Nukleare Räume aus M. K.-Funktionen, Math. Z. 92 (1966), 295–306.
A. C. Zaanen, Integration, North-Holland Publishing Company, Amsterdam (1967).
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Reiher, K. Weighted inductive and projective limits of normed Köthe function spaces. Results. Math. 13, 147–161 (1988). https://doi.org/10.1007/BF03323402
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DOI: https://doi.org/10.1007/BF03323402