1989
DOI: 10.1090/s0002-9939-1989-0933516-5
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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 co… Show more

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Cited by 5 publications
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“…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.…”
Section: O( a D) As Frechet Spaces If And Only If M Is Hyperconvexmentioning
confidence: 99%
“…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.…”
Section: O( a D) As Frechet Spaces If And Only If M Is Hyperconvexmentioning
confidence: 99%
“…
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.

…”
mentioning
confidence: 99%
“…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.…”
mentioning
confidence: 99%