Some Invariants of Frechet Spaces and Imbeddings of Smooth Sequence Spaces

Abstract

IntroductionThe purpose of this article is to give an exposition of some recent results related to imbedding smooth sequence spaces into nuclear Frechet spaces. Although some new results are stated and proved in the present article, essentially they are modifications of the results of Aytuna, Krone and the author which are contained in [6] and [7].

Multirectangular invariants for power Köthe spaces

“…Important instruments in those investigations are classical linear topological invariants (approximative and diametrical dimensions, see, e.g., [2,16,20,24]). These invariants are used at their best for regular Köthe spaces [10,11,13,14,17,25,27], in particular, for the spaces (1) with a i ↑ ∞. Problem 1 for nonMontel (a i → ∞) spaces (1), which turns to be beyond powers of the classical invariants, was investigated in [21][22][23] (with l 2 -norms instead of l 1 -norms) by means of some new invariants based on spectral behavior of the operator generating the scale; these invariants exerted an influence on further development of linear topological invariants dealing with non-regular spaces, especially on its early stage).…”

“…Now we show that the construction of synthetic neighborhoodsŨ andṼ provides the estimate (18). Due to (25), it is sufficient to check the estimate…”

Some Remarks on Diametral Dimension and Approximate Diametral Dimension of Certain Nuclear Fréchet Spaces

“…Linear topological invariants DN and Ω are enjoyed by many natural nuclear Fréchet spaces appearing in analysis. In particular, spaces of analytic functions, solutions of homogeneous elliptic linear partial differential operators with their natural topologies have the properties DN and Ω, see [13] and [15].…”

Advances in the Theory of Fréchet Spaces

“…, then E and Aoo(a) are in fact isomorphic. For more information on this and for diverse applications of this result we refer the reader to the reports of Terzioglu[100] and Krone On the other hand 1.1 Proposition (3) says that there is an imbedding of O(C d ) into OeM) as a closed subspace.…”