On the existence of Schauder bases in Sobolev spaces
On the Fejér means of bounded Ciesielski systems
We investigate the bounded Ciesielski systems, which can be obtained from the spline systems of order (m,k) in the same way as the Walsh system arises from the Haar system. It is shown that the maximal operator of the Fejér means of the Ciesielski-Fourier series is bounded from the Hardy space to if 1/2 < p < ∞ and m ≥ 0, |k| ≤ m + 1. Moreover, it is of weak type (1,1). As a consequence, the Fejér means of the Ciesielski-Fourier series of a function f converges to f a.e. if f ∈ L₁ as n...
On the Gelfand-Phillips property in Banach spaces with PRI.
On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems
In one of the earliest monographs that involve the notion of a Schauder basis, Franklin showed that the Gram-Schmidt orthonormalization of a certain Schauder basis for the Banach space of functions continuous on [0,1] is again a Schauder basis for that space. Subsequently, Ciesielski observed that the Gram-Schmidt orthonormalization of any Schauder system is a Schauder basis not only for C[0,1], but also for each of the spaces , 1 ≤ p < ∞. Although perhaps not probable, the latter result would...
On the non-equivalence of rearranged Walsh and trigonometric systems in
We consider the question of whether the trigonometric system can be equivalent to some rearrangement of the Walsh system in for some p ≠ 2. We show that this question is closely related to a combinatorial problem. This enables us to prove non-equivalence for a number of rearrangements. Previously this was known for the Walsh-Paley order only.
On the relation between several notions of unconditional structure - A counterexample
On the representation of uncountable symmetric basic sets and its applications
It is shown that every uncountable symmetric basic set in an F-space with a symmetric basis is equivalent to a basic set generated by one vector. We apply this result to investigate the structure of uncountable symmetric basic sets in Orlicz and Lorentz spaces.
On the Riesz-Fischer theorem in a smooth Banach space
On the structure of Banach spaces with an unconditional basic sequence
For a Banach space X with an unconditional basic sequence, one of the following regular-irregular alternatives holds: either X contains a subspace isomorphic to ℓ₂, or X contains a subspace which has an unconditional finite-dimensional decomposition, but does not admit such a decomposition with a uniform bound for the dimensions of the decomposition. This result can be viewed in the context of Gowers' dichotomy theorem.
On the Surjective Dunford-Pettis Property.
On the surjective Dunford-Pettis property
On the type constants with respect to systems of characters of a compact abelian group
We prove that there exists an absolute constant c such that for any positive integer n and any system Φ of characters of a compact abelian group, , where T is an arbitrary operator between Banach spaces, is the type norm of T with respect to Φ and is the usual Rademacher type-2 norm computed with n vectors. For the system of the first Walsh functions this is even true with c=1. This result combined with known properties of such type norms provides easy access to quantitative versions of...
On the volume method in the study of Auerbach bases of finite-dimensional normed spaces
In this note we show that if the ratio of the minimal volume V of n-dimensional parallelepipeds containing the unit ball of an n-dimensional real normed space X to the maximal volume v of n-dimensional crosspolytopes inscribed in this ball is equal to n!, then the relation of orthogonality in X is symmetric. Hence we deduce the following properties: (i) if V/v=n! and if n>2, then X is an inner product space; (ii) in every finite-dimensional normed space there exist at least two different Auerbach...
On the volume of unit balls in Banach spaces
On the weak-basis theorem
On three problems from the Scottish Book connected with orthogonal systems
On Two Classes of Mappings on Banach Sequenee Spaces.
On unconditional convergence of series in Banach lattices
On unconditional polynomial bases of the space
On unconditionally saturated Banach spaces
We prove a structural property of the class of unconditionally saturated separable Banach spaces. We show, in particular, that for every analytic set 𝓐, in the Effros-Borel space of subspaces of C[0,1], of unconditionally saturated separable Banach spaces, there exists an unconditionally saturated Banach space Y, with a Schauder basis, that contains isomorphic copies of every space X in the class 𝓐.