Quasinormability of sorne spaces of holornorphic mappings
نویسنده
چکیده
A class of locally convex vector spaces with a special Schauder decomposition is considered. It is proved that the elements of this class, which includes sorne spaces naturally appearing in infinite dimensional holomorphy, are quasinormable though ja general they are neither metrizable nor Schwartz spaces. O. INTRODUCTION AND PRELIMINARIES Let X be a ¡Iausdorff real or complex loeally convex vector space, and let j3(X’, X) denote tite strong topology on tite dual X’ of X. In 1954, Grotitendieck [3] proved tite following result. Theorem O. For any Hausdorff lx. vector space X, ihe following assertions are equivalení: (a) for any equicontinuous subseí E of X’ ihere isa neighbourhood Vof O ¡ti X such thai 13(10, X) induces on e ihe topology of unform con vergence over V. (Li) For any neighbourhood U of O in X diere Ls a neighbourhood y of O in X such thai for any a >0 diere is a bounded subseí M ¡ti X with VGM+aU Locally convex spaces for which titese two statements hoid are said to be quasinormable, and this class of spaces was studied in [3]. Most of the spaces in Functional Analysis belong to titis class or have a close relationsbip with it. On the otiter hand, titere are spaces (even Fréehet spaces) which are not quasinormable, tite first example of such a space is dueto Kóthe in 1948. (See [4], 10.7). () Supported by the D.G.í.C.Y.T., June 14. 1988. 1980 Mathematics Subject Classification (1985 revision): 46020. Editorial de la Universidad Complutense. Madrid 1990. http://dx.doi.org/10.5209/rev_REMA.1990.v3.n1.18033
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تاریخ انتشار 2014