The existence of bases in complemented nuclear subspaces of infinite type power series spaces

Abstract

The problem of existence of bases in an arbitrary complemented subspace of an infinite type nuclear power series space, posed by B. S. Mityagin, was solved in [1,3] only under various additional restrictions on the space or on the complemented subspace.Some specific infinite type power series spaces are widely used in research. These are the space of rapidly decreasing functions (or its realization as the space s of rapidly decreasing sequences), spaces of entire functions with the topology of uniform convergence on compact sets, and others.The KSthe space]~n ]P e-x~ rpb,~ =]~lr <*c, rEN , 1 <~p~< oc, of number sequences equipped with the topology defined by the system of norms (I " It) is called an infinite type power series space.The study of bases and structural properties of infinite type power series spaces has led to the discovery of linear topological invariants, namely, the geometric properties D1 and ~, shared by all these spaces and their complemented subspaces [4][5][6].Let E be an infinite type nuclear power series space isomorphic to the Cartesian square of itself. Then necessary and sufficient conditions are known for a Fr~chet space with properties D1 and gt to be isomorphic to a complemented subspace in E (see [6]). These conditions are expressed in terms of the asymptotics of n:widths of neighborhoods of zero (the diametral dimension).In the following, we give a general result that implies a complete solution of the problem of existence of bases in an arbitrarily complemented subspace of the space s = /2[n ~] of rapidly decreasing sequences as well as of the problem of characterizing complemented subspaces of infinite t.~e nuclear power series spaces. In particular, we strengthen the above-mentioned result of [6] by removing the requirement that E be isomorphic to the Cartesian square of E.Let us recall the definitions of the geometric properties D1 and ft. Definition 1 [4,5]. A Frdchet space (E, ([ 9 It)) is said to have property D1 (sometimes denoted by DN) if the following condition holds: there exists a continuous norm tl " I] such that for all r E N there exist s(r) E N and c(r) > 0 such that I" I~ < e(r)ll" Ill" I~(~>.Let {Ur} be a base of closed absolutely convex neighborhoods of zero in a Fr6chet space E, that is, Ur = {e e E : lelr ~< 1}, r e N.