Class Of Separable Banach Spaces Research Articles

Asymptotic smoothness and universality in Banach spaces

For $1 \lt p\leq \infty $, we study the complexity and the existence of universal spaces for two classes of separable Banach spaces, denoted $\mathsf{A}_p$ and $\mathsf{N}_p$, and related to asymptotic smoothness in Banach spaces. We show that each of the

A Family of Banach Spaces Over ℝ∞

In T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016), the topology of [Formula: see text] was replaced with a new topology and denoted by [Formula: see text]. This space was then used to construct Lebesgue measure on [Formula: see text] in a manner that is no more difficult than the same construction on [Formula: see text]. More important for us, a new class of separable Banach spaces [Formula: see text], [Formula: see text], for the HK-integrable functions, was introduced. These spaces also contain the [Formula: see text] spaces and the Schwartz space as continuous dense embeddings. This paper extends the work in T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016) from [Formula: see text] to [Formula: see text].

Garling sequence spaces

The aim of this paper is to introduce and investigate a new class of separable Banach spaces modeled after an example of Garling from 1968. For each $1\leqslant p<\infty$ and each nonincreasing weight $\textbf{w}\in c_0\setminus\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous, and uniformly subprojective Banach space $g(\textbf{w},p)$. We also show that $g(\textbf{w},p)$ admits a unique subsymmetric basis despite the fact that for a wide class of weights it does not admit a symmetric basis. This provides the first known examples of Banach spaces where those two properties coexist.

The metric geometry of the Hamming cube and applications

The Lipschitz geometry of segments of the infinite Hamming cube is studied. Tight estimates on the distortion necessary to embed the segments into spaces of continuous functions on countable compact metric spaces are given. As an application, the first nontrivial lower bounds on the $C(K)$-distortion of important classes of separable Banach spaces, where $K$ is a countable compact space in the family $ \{ [0,\omega],[0,\omega\cdot 2],\dots, [0,\omega^2], \dots, [0,\omega^k\cdot n],\dots,[0,\omega^\omega]\}\ ,$ are obtained.

The fixed point property and the Opial condition on tree-like Banach spaces

We introduce some new tree-like Banach spaces, belonging to the class of separable Banach spaces not containing $\ell_1$ with non-separable dual, each one of which satisfies the following: $(1)$~the space has the fixed point property and $(2)$~the space does not satisfy the Opial condition. In addition, one of these spaces contains subspaces isomorphic to $c_0$, whose Banach-Mazur distance from $c_0$ becomes arbitrarily large.

On the complexity of some inevitable classes of separable Banach spaces

In this paper, we study the descriptive complexity of some inevitable classes of Banach spaces. Precisely, as shown in [11], every Banach space either contains a hereditarily indecomposable subspace or an unconditional basis, and, as shown in [8], every Banach space either contains a minimal subspace or a continuously tight subspace. In this note, we study the complexity of those inevitable classes as well as the complexity of containing a subspace in any of those classes.

On the complexity of some classes of Banach spaces and non-universality

These notes are dedicated to the study of the complexity of several classes of separable Banach spaces. We compute the complexity of the Banach-Saks property, the alternating Banach-Saks property, the complete continuous property, and the LUST property. We also show that the weak Banach-Saks property, the Schur property, the Dunford-Pettis property, the analytic Radon-Nikodym property, the set of Banach spaces whose set of unconditionally converging operators is complemented in its bounded oper- ators, the set of Banach spaces whose set of weakly compact operators is complemented in its bounded operators, and the set of Banach spaces whose set of Banach-Saks opera- tors is complemented in its bounded operators, are all non Borel in SB. At last, we give several applications of those results to non-universality results.

Estimation of the Szlenk index of Banach spaces via Schreier spaces

For each ordinal $\alpha <\omega _1$, we prove the existence of a Banach space with a basis and Szlenk index $\omega ^{\alpha +1}$ which is universal for the class of separable Banach spaces with Szlenk index not exceeding $\omega ^\alpha $. Our proof inv

On classes of Banach spaces admitting ‘‘small” universal spaces

We characterize those classes C of separable Banach spaces admitting a separable universal space Y (that is, a space Y containing, up to isomorphism, all members of C) which is not universal for all separable Banach spaces. The characterization is a byproduct of the fact, proved in the paper, that the class NU of non-universal separable Banach spaces is strongly bounded. This settles in the affirmative the main conjecture from Argyros and Dodos (2007). Our approach is based, among others, on a construction of L ∞ -spaces, due to J. Bourgain and G. Pisier. As a consequence we show that there exists a family {Y ξ : ξ < ω 1 } of separable, non-universal, L ∞ -spaces which uniformly exhausts all separable Banach spaces. A number of other natural classes of separable Banach spaces are shown to be strongly bounded as well.

A new class of Banach spaces

In this paper, we construct a new class of separable Banach spaces , for 1 ⩽ p ⩽ ∞, each of which contains all of the standard Lp spaces, as well as the space of finitely additive measures, as compact dense embeddings. Equally important is the fact that these spaces contain all Henstock–Kurzweil integrable functions and, in particular, the Feynman kernel and the Dirac measure, as norm bounded elements. As a first application, we construct the elementary path integral in the manner originally intended by Feynman. We then suggest that is a more appropriate Hilbert space for quantum theory, in that it satisfies the requirements for the Feynman, Heisenberg and Schrödinger representations, while the conventional choice only satisfies the requirements for the Heisenberg and Schrödinger representations. As a second application, we show that the mixed topology on the space of bounded continuous functions, , used to define the weak generator for a semigroup T(t), is stronger than the norm topology on . (This means that, when extended to is strongly continuous, so that the weak generator on becomes a strong generator on .)

A Dolbeault isomorphism theorem in infinite dimensions

For a large class of separable Banach spaces, we prove the real analytic Dolbeault isomorphism theorem for open subsets.

Approximation by smooth functions with no critical points on separable Banach spaces

We characterize the class of separable Banach spaces X such that for every continuous function f : X → R and for every continuous function ε : X → ( 0 , + ∞ ) there exists a C 1 smooth function g : X → R for which | f ( x ) − g ( x ) | ⩽ ε ( x ) and g ′ ( x ) ≠ 0 for all x ∈ X (that is, g has no critical points), as those infinite-dimensional Banach spaces X with separable dual X ∗ . We also state sufficient conditions on a separable Banach space so that the function g can be taken to be of class C p , for p = 1 , 2 , … , + ∞ . In particular, we obtain the optimal order of smoothness of the approximating functions with no critical points on the classical spaces ℓ p ( N ) and L p ( R n ) . Some important consequences of the above results are (1) the existence of a non-linear Hahn–Banach theorem and the smooth approximation of closed sets, on the classes of spaces considered above; and (2) versions of all these results for a wide class of infinite-dimensional Banach manifolds.

Genericity and amalgamation of classes of Banach spaces

We study universality problems in Banach space theory. We show that if A is an analytic class, in the Effros–Borel structure of subspaces of C ( [ 0 , 1 ] ) , of non-universal separable Banach spaces, then there exists a non-universal separable Banach space Y, with a Schauder basis, that contains isomorphs of each member of A with the bounded approximation property. The proof is based on the amalgamation technique of a class C of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space R not containing L 1 ( 0 , 1 ) such that the indices β and r N D are unbounded on the set of Baire-1 elements of the ball of the double dual R ∗ ∗ of R. This answers two questions of H.P. Rosenthal. We also introduce the concept of a strongly bounded class of separable Banach spaces. A class C of separable Banach spaces is strongly bounded if for every analytic subset A of C there exists Y ∈ C that contains all members of A up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space X.

Boundedly complete basic sequences,c 0-subspaces, and injections of Banach spaces

We study the connection between topological properties of subsets of a given Banach space and their images under linear, continuous one-to-one mappings on the one hand and the existence in a given Banach space of either a boundedly complete basic sequence (BCBS) or an isomorphic copy ofco (co-subspace) on the other hand. We present criteria for the existence of a BCBS. They are deduced from new characterisations ofGδ-embeddings which we also present. We obtain a necessary and sufficient condition for separability of a dual Banach space in terms of saturation by BCBS. Criteria for the existence in a Banach space of aco-subspace are also presented. We describe the class of separable Banach spaces which contains either a BCBS or aco-subspace.

Separable Banach spaces which admitl n ∞ approximations

In this paper we study a class of separable Banach spaces which can be approximated by certain special finite-dimensional subspaces. This class is characterized in Theorem 1.1, from which it follows that the space of continuous scalar-valued functions on a compact metric space always belongs to this class, and that every member of this class has a monotone basis.