VMO spaces not having Schauder basis

ПустьA _n (n=0, 1, 2, ...) — последо вательность δ-алгебр, порождаемых двоичны ми отрезками [k2^?n , (k+1)2^?n ) (0≦k<2 ⁿ ). Замыкание множества двоичных с тупенчатых функций по норме      

Norm-one complemented hyperplanes in spaces with a Schauder basis

Some results on existence of norm one projections onto hyperplanes in spaces with a Schauder basis are given. Possible characterizations of Hilbert spaces using this property are also discussed.     

A geometrical characterization of strongly regular graphs

We give a necessary and sufficient condition of a Euclidean representation of a simple graph to be spherical. Moreover we show a characterization of strongly regular graphs from the view point of Euclidean representations of a graph. From this characterization, we define a natural generalized concept of a strongly regular graph closely related to Euclidean designs and codes.     

A characterization of geometrical mappings of Grassmann spaces

In the paper we show that mappings of Grassmann spaces sending base subsets to base subsets are induced by strong embeddings of the corresponding projective spaces.     

On compactoidity in non-archimedean locally convex spaces with a Schauder basis

Several properties of compactoid sets in non-archimedean locally convex spaces with a Schauder basis are proved in this paper. As a consequence we derive partial affirmative answers to the questions formulated by Gruson and Van der Put ([4], problem following 5.8) and Schikhof ([6], problem following 1.7), respectively.     

Gaps and Fatou theorems for series in Schauder basis of holomorphic functions

We prove a gap theorem and the “Fatou change-of-sign theorem” [Fatou, P., 1906, Sèries trigonométriques e séries de Taylor. Acta Mathematica, 39, 335–400] for expansions in common Schauder basis of holomorphic functions.     

Examples of non-archimedean nuclear Frécheat spaces without a Schauder basis

We solve the problem of the existence of a Schauder basis in non-archimedean Fréchet spaces of countable type (stated in [3]). Using examples of real nuclear Fréchet spaces without a Schauder basis (of Bessaga [1]), Mitiagin [5] and Vogt [10]) we construct examples of non-archimedean nuclear Fréchet spaces without a Schauder basis (even without the bounded approximation property).     

A characterization of exponential and geometrical distributions and its stability bound

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, pp. 68–72, 1986.     

A separable space with no Schauder decomposition

We combine some known results to remark that there exists a separable Banach space which fails to have a Schauder decomposition. It can be chosen as a subspace of Gowers-Maurey space without any unconditional basic sequence.

     

A generalization of the Leray—Schauder index formula

This paper generalizes the Leray-Schauder index formula to the case where the inverse image of a point consists of a smooth manifold, assuming some nondegeneracy condition is satisfied on the manifold. The result states that the index is the Euler characteristic of a certain vector bundle over the manifold. Under slightly stronger nondegeneracy conditions, the index is in fact the Euler characteristic of the manifold. The paper also includes a discussion of the Euler characteristic for vector bundles and a simple proof of the Gauss-Bonnet-Chern theorem.     

Failure rate and characterization of the geometrical distribution

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei —Trudy Seminara, pp. 112–117, 1985.     

A note on cones associated to Schauder bases

We study the existence of infima of subsets in Banach spaces ordered by normal cones associated to shrinking Schauder bases. Under these conditions we prove the existence of infima for a class of subsets verifying a weakly compactness property. Moreover we prove that a normal cone associated to a Schauder basis in a reflexive Banach space is strongly minihedral extending the known result for unconditional Schauder bases. Several examples are also discussed. Miguel Sama: The work of this author is partially supported by Ministerio de Educación y Ciencia (Spain), project MTM2006-02629 and Ingenio Mathematica (i-MATH) CSD2006-00032 (ConsoliderIngenio 2010).     

A Characterization of Schauder Frames Which Are Near-Schauder Bases

A basic problem of interest in connection with the study of Schauder frames in Banach spaces is that of characterizing those Schauder frames which can essentially be regarded as Schauder bases. In this paper, we give a solution to this problem using the notion of the minimal-associated sequence spaces and the minimal-associated reconstruction operators for Schauder frames. We prove that a Schauder frame is a near-Schauder basis if and only if the kernel of the minimal-associated reconstruction operator contains no copy of c ₀. In particular, a Schauder frame of a Banach space with no copy of c ₀ is a near-Schauder basis if and only if the minimal-associated sequence space contains no copy of c ₀. In these cases, the minimal-associated reconstruction operator has a finite dimensional kernel and the dimension of the kernel is exactly the excess of the near-Schauder basis. Using these results, we make related applications on Besselian frames and near-Riesz bases.