Characterizations of pairs (E,F) of complete (LF)?spaces such that every continuous linear map from E to F maps a 0?neighbourhood of E into a bounded subset of F are given. The case of sequence (LF)?spaces is also considered. These results are similar to the ones due to D. Vogt in the case E and F are Fréchet spaces. The research continues work of J. Bonet, A. Galbis, S. Önal, T. Terzioglu and D. Vogt.
It is proved that a Fréchet space is quasinormable if, and only if, every quotient space satisfies the density condition of Heinrich. This answers positively a conjecture of Bonet and Díaz
We give a representation of the spaces as spaces of vector-valued sequences and use it to investigate their topological properties and isomorphic classification. In particular, it is proved that is isomorphic to the sequence space , thereby showing that the isomorphy class does not depend on the dimension N if p=2.
We prove that the direct sum and the product of countably many copies of L[0, 1] are primary locally convex spaces. We also give some related results.
We characterize Köthe echelon spaces (and, more generally, those Fréchet spaces with an unconditional basis) which are Schwartz, in terms of the convergence of the Cesàro means of power bounded operators defined on them. This complements similar known characterizations of reflexive and of Fréchet-Montel spaces with a basis. Every strongly convergent sequence of continuous linear operators on a Fréchet-Schwartz space does so in a special way. We single out this type of "rapid convergence" for a sequence...
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