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, Volume 2, Issue 4, pp 359–364 | Cite as

Über kompakte Operatoren in lokalkonvexen Räumen

  • Ulrich Mertins


Let E, F be (B)-spaces and let E have the approximation property. Then the closure of F′⊗E in Lb (F,E) is identical with the space K(F,E) of compact maps in L(F,E). The following generalization of this well known statement is proved: Let E, F be complete, locally convex spaces and let E have the approximation property. Then
, where
denotes a 0-neighborhood base in F and
refers to the completion of the tensor product with respect to the topology of bi-equicontinuous convergence.

The last part is concerned with the adjoint map of a compact operator.


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    GROTHENDIECK, A.: Produits tensoriels topologiques et espaces nucléaires. Mem.Amer.math.Soc. 16 1955.Google Scholar
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    KÖTHE, G.: Zur Theorie der kompakten Operatoren in lokalkonvexen Räumen. Portugaliae Math. 13 97–104 (1954)Google Scholar
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    ———: Topologische lineare Räume, 2.Aufl. Berlin-Heidelberg-New York: Springer 1966.Google Scholar
  4. [4]
    SCHAEFER, H.H.: Topological vector spaces, New York: Macmillan 1966.Google Scholar

Copyright information

© Springer-Verlag 1970

Authors and Affiliations

  • Ulrich Mertins
    • 1
  1. 1.Mathematisches InstitutUniversität Karlsruhe (TH)75 Karlsruhe 1

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