In this work, we are dealing with the natural topology in a perfect sequence space and the transfert of topologies of a locally K− convex space E with a Schauder basis (e i ) i to such Space. We are also interested with the compatible topologies on E for which the basis(e i ) i is equicontinuous, and the weak basis problem. Finally, we give some applications to barrelled Spaces and G−Spaces.

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M. sova [10] proved that the infinitesimal generator of all uniformly continuous cosine family, of operators in Banach space, is a bounded operator. We show by counter-example that the result mentioned above is not true in general on Fréchet spaces, and we prove that the infinitesimal generator of an uniformly continuous cosine family of operators in a class of Fréchet spaces (quojection) is necessarily continuous.

The purpose of the present paper is to develop a theory of a duality in sequence spaces over a non-archimedean vector space. We introduce polar topologies in such spaces, and we give basic results characterizing compact, C-compact, complete and AK−complete subsets related to these topologies.

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