On projections in power series spaces and the existence of bases

Abstract

Abstract.Mityagin posed the problem, whether complemented subspaces of nuclear infinite type power series spaces have a basis. A related more general question was asked by Pelczynski. It is well known for a complemented subspace £ of a nuclear infinite type power series space, that its diametral dimension can be represented by AE = AAoo(q) for a suitable sequence a with a, > \n(j + 1). In this article we prove the existence of a basis for E in case that aj > j and sup -^ < oo .It was shown by Mityagin, that complemented subspaces of nuclear finite type power series spaces always have a basis, and he asked, whether the same is valid for infinite type (cf. [3,4,5]). Dubinsky and Vogt [2] obtained a positive solution for some nuclear power series spaces Aoo(a), namely they assumed that the set of all finite limit points of {^-: i, j € N} is bounded. Results for some other special cases are stated below. Pelczynski [6] posed the more general problem, whether complemented subspaces of nuclear Fréchet spaces with basis again have a basis. Both problems are open up to now.Every complemented subspace E of a nuclear infinite type power series space has the same diametral dimension as a power series space A^ja) for a suitable sequence a with a >ln(y'-f-l) (cf. Terzioglu [8]). Considering isomorphisms between spaces of analytic functions Zaharjuta [15] conjectured that E has a basis for stable a (this means sup ^ < oo ). This will be proved in the present note in case a > /. Other positive solutions have been obtained if one of the three following assumptions is satisfied:(1) Aoo(a) is a complemented subspace of E (cf. Vogt [10]).(2) There is a tame projection onto E (cf. Dubinsky/Vogt [2]). (3) E is isomorphic to E ® E (cf. Wagner [14], see also [11]). The present proof uses result no. 1 of Vogt [10]. The main tool is the construction of a basis and of a projection in E by a permuted Gram-Schmidt orthonormalization.

Advances in the Theory of Fréchet Spaces

“…then one has a control on the norms and the dual norms of the common orthonormal basis of Hoo and Ho and indeed under these assumptions it follows that X can be presented as Q(Hoo; Ho) [108]. (For other versions of this method see [65], [37, 2.2. (For other versions of this method see [65], [37, 2.2.…”

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

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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.

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“…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.…”