On the complemented subspaces of the Schreier spaces

Illinois Journal of Mathematics, 2021

For each countable ordinal α let S α be the Schreier set of order α and X Sα be the corresponding Schreier space of order α. In this paper we prove several new properties of these spaces. (1) If α is non-zero then X Sα possesses the λ-property of R. Aron and R. Lohman and is a (V)-polyhedral spaces in the sense on V. Fonf and L. Vesely. (2) If α is non-zero and 1 < p < ∞ then the p-convexification X p Sα possesses the uniform λ-property of R. Aron and R. Lohman. (3) For each countable ordinal α the space X * Sα has the λ-property. (4) For n ∈ N, if U : X Sn → X Sn is an onto linear isometry then U e i = ±e i for each i ∈ N. Consequently, these spaces are light in the sense of Megrelishvili. The fact that for non-zero α, X Sα is (V)-polyhedral and has the λ-property implies that each X Sα is an example of space solving a problem of J. Lindenstrauss from 1966. The first example of such a space was given by C. De Bernardi in 2017 using a renorming of c 0 .

Israel Journal of Mathematics, 1991

In this paper we prove some results related to the problem of isomorphically classifying the complemented subspaces of X p. We characterize the complemented subspaces of X p which are isomorphic to X p by showing that such a space must contain a canonical complemented subspace isomorphic to X p. We also give some characterizations of complemented subspaces of X p isomorphic to ℓ p ⊕ ℓ 2 .

Annali di Matematica Pura ed Applicata, 2001

On Complemented Subspaees of the Spaces (lP)NN lq(l q) (*).

JOURNAL OF OPERATOR THEORY

arXiv (Cornell University), 2023

In this paper, we prove the following results. There exists a Banach space without basis which has a Schauder frame. There exists an universal Banach space B (resp. B) with a basis (resp. an unconditional basis) such that, a Banach X has a Schauder frame (resp. an unconditional Schauder frame) if and only if X is isomorphic to a complemented subspace of B (resp.B). For a weakly sequentially complete Banach space, a Schauder frame is unconditional if and only if it is besselian. A separable Banach space X has a Schauder frame if and only if it has the bounded approximation property. Consequenty, The Banach space L(H, H) of all bounded linear operators on a Hilbert space H has no Schauder frame. Also, if X and Y are Banach spaces with Schauder frames then, the Banach space X ⊗ π Y (the projective tensor product of X and Y) has a Schauder frame. From the Faber−Schauder system we construct a Schauder frame for the Banach space C[0, 1] (the Banach space of continuous functions on the closed interval [0, 1]) which is not a Schauder basis of C[0, 1]. Finally, we give a positive answer to some open problems related to the Schauder bases (In the Schauder frames setting).

Symmetry

A Schauder basis in a real or complex Banach space X is a sequence ( e n ) n ∈ N in X such that for every x ∈ X there exists a unique sequence of scalars ( λ n ) n ∈ N satisfying that x = ∑ n = 1 ∞ λ n e n . Schauder bases were first introduced in the setting of real or complex Banach spaces but they have been transported to the scope of real or complex Hausdorff locally convex topological vector spaces. In this manuscript, we extend them to the setting of topological vector spaces over an absolutely valued division ring by redefining them as pre-Schauder bases. We first prove that, if a topological vector space admits a pre-Schauder basis, then the linear span of the basis is Hausdorff and the series linear span of the basis minus the linear span contains the intersection of all neighborhoods of 0. As a consequence, we conclude that the coefficient functionals are continuous if and only if the canonical projections are also continuous (this is a trivial fact in normed spaces but no...

Proceedings of the American Mathematical Society, 2005

Topology and its Applications, 2006

We introduce notions of nearly good relations and N-sticky modulo a relation as tools for proving that spaces are D-spaces. As a corollary to general results about such relations, we show that Cp(X) is hereditarily a D-space whenever X is a Lindelöf Σ-space. This answers a question of Matveev, and improves a result of Buzyakova, who proved the same result for X compact. We also prove that if a space X is the union of finitely many D-spaces, and has countable extent, then X is linearly Lindelöf. It follows that if X is in addition countably compact, then X must be compact. We also show that Corson compact spaces are hereditarily D-spaces. These last two results answer recent questions of Arhangel'skii. Finally, we answer a question of van Douwen by showing that a perfectly normal collectionwise-normal non-paracompact space constructed by R. Pol is a D-space. 1991 Mathematics Subject Classification. 54D20. I would like to dedicate this paper to my colleague and friend A.V. Arhangel'skii on the occasion of his 65 th birthday.

Proceedings of the American Mathematical Society, 2021

The λ-property and the uniform λ-property were first introduced by R. Aron and R. Lohman in 1987 as geometric properties of Banach spaces. In 1989, Th. Shura and D. Trautman showed that the Schreier space posseses the λ-property and asked if it has the uniform λ-property. In this paper, we show that Schreier space does not have the uniform λ-property. Furthermore, we show that the dual of the Schreier space does not have the uniform λ-property.

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.