Characterization of inner product spaces in V- and L-spaces

In [4], Freese and Murphy introduce a new class of spaces, the V-spaces, which include Banach spaces, hyperbolic spaces, and other metric spaces. In this class of spaces they investigate conditions which are equivalent to strict convexity in Banach spaces, and extend some of the Banach space results to this new class of spaces. It is natural to ask if known characterizations of real inner product spaces among Banach spaces can also be extended to this larger class of spaces. In the present paper it will be shown that a metrization of an angle bisector property used in [3] to characterize real inner product spaces among Banach spaces also characterizes real inner product spaces among V-spaces, and among another class of spaces, the L-spaces, which include hyperbolic spaces and strictly convex Banach spaces. In the process it is shown that in a complete, convex, externally convex metric space M, if the foot of a point on a metric line is unique, then M satisfies the monotone property, thus answering a question raised in [4].     

Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces

In this paper, the class of nonspreading mappings in Banach spaces is introduced. This class contains the recently introduced class of firmly nonexpansive type mappings in Banach spaces and the class of firmly nonexpansive mappings in Hilbert spaces. Among other things, we obtain a fixed point theorem for a single nonspreading mapping in Banach spaces. Using this result, we also obtain a common fixed point theorem for a commutative family of nonspreading mappings in Banach spaces. Received: 10 August 2007     

A Class of Metric Spaces Which Do Not Coarsely Contain Expanders

In this paper, a class of metric spaces which include Hilbert spaces and Hadamard manifolds are defined. And the expanders cannot be coarsely embedded into this class of metric spaces are proved.     

Weak Similarities of Finite Ultrametric and Semimetric Spaces

Weak similarities form a special class of mappings between semimetric spaces. Two semimetric spaces X and Y are weakly similar if there exists a weak similarity Φ: X → Y. We find a structural characteristic of finite ultrametric spaces for which the isomorphism of its representing trees implies a weak similarity of the spaces. We also find conditions under which the Hasse diagrams of balleans of finite semimetric spaces are isomorphic.     

Positive definite metric spaces

Magnitude is a numerical invariant of finite metric spaces, recently introduced by Leinster, which is analogous in precise senses to the cardinality of finite sets or the Euler characteristic of topological spaces. It has been extended to infinite metric spaces in several a priori distinct ways. This paper develops the theory of a class of metric spaces, positive definite metric spaces, for which magnitude is more tractable than in general. Positive definiteness is a generalization of the classical property of negative type for a metric space, which is known to hold for many interesting classes of spaces. It is proved that all the proposed definitions of magnitude coincide for compact positive definite metric spaces and further results are proved about the behavior of magnitude as a function of such spaces. Finally, some facts about the magnitude of compact subsets of $\ell _p^n$ for $p \le 2$ are proved, generalizing results of Leinster for $p=1,2$ using properties of these spaces which are somewhat stronger than positive definiteness.     

Maximally movable spaces of Finsler type and their generalization

In this paper, we consider some generalization of maximally movable spaces of Finsler type. Among them, there are locally conic spaces (Riemannian metrics of their tangent spaces are realized on circular cones) and generalized Lagrange spaces with Tamm metrics (their tangent Riemannian spaces admit all rotations). On the tangent bundle of a Riemannian manifold, we study a special class of almost product metrics, generated Tamm metric. This class contains Sasaki metric and Cheeger–Gromol metric. We determine the position of this class in the Naveira classification of Riemannian almost product metrics.     

Grothendieck space ideals and weak continuity of polynomials on locally convex spaces

We introduce the classes of locally convex spaces with the local Dunford-Pettis property and locally dual Schur spaces. We examine their properties and their relationship to other classes of locally convex spaces. In the class of locally convex spaces with the local Dunford-Pettis property all polynomials are weakly sequentially continuous whereas in the class of locally dual Schur spaces all polynomials are weakly continuous on bounded sets.     

Sperner spaces and first‐order logic

We study the class of Sperner spaces, a generalized version of affine spaces, as defined in the language of pointline incidence and line parallelity. We show that, although the class of Sperner spaces is a pseudo‐elementary class, it is not elementary nor even ??_∞ω‐axiomatizable. We also axiomatize the first‐order theory of this class.     

Which spaces for design?

We determine the largest class of spaces of sufficient regularity which are suitable for design in the sense that they do possess blossoms. It is the class of all spaces containing constants of which the spaces derived under differentiation are Quasi Extended Chebyshev spaces, i.e., they permit Hermite interpolation, Taylor interpolation excepted. It is also the class of all spaces which possess Bernstein bases, or of all spaces for which any associated spline space does possess a B-spline basis. Note that blossoms guarantee that such bases are normalised totally positive bases. They even are the optimal ones.     

Triplets of Closely Embedded Dirichlet Type Spaces on the Unit Polydisc

We propose a general concept of triplet of Hilbert spaces with closed embeddings, instead of continuous ones, and we show how rather general weighted $L^2$ spaces yield this kind of generalized triplets of Hilbert spaces for which the underlying spaces and operators can be explicitly calculated. Then we show that generalized triplets of Hilbert spaces with closed embeddings can be naturally associated to any pair of Dirichlet type spaces $\mathcal{D }_\alpha (\mathbb{D }^N)$ of holomorphic functions on the unit polydisc $\mathbb{D }^N$ and we explicitly calculate the associated operators in terms of reproducing kernels and radial derivative operators. We also point out a rigging of the Hardy space $H^2(\mathbb{D }^N)$ through a scale of Dirichlet type spaces and Bergman type spaces.     

Grothendieck space ideals and weak continuity of polynomials on locally convex spaces

We introduce the classes of locally convex spaces with the local Dunford-Pettis property and locally dual Schur spaces. We examine their properties and their relationship to other classes of locally convex spaces. In the class of locally convex spaces with the local Dunford-Pettis property all polynomials are weakly sequentially continuous whereas in the class of locally dual Schur spaces all polynomials are weakly continuous on bounded sets. Research supported by Science Foundation Ireland, Basic Research Grant 2004.     

Interpolation and Embeddings of Weighted Tent Spaces

Given a metric measure space X, we consider a scale of function spaces \(T^{p,q}_s(X)\), called the weighted tent space scale. This is an extension of the tent space scale of Coifman, Meyer, and Stein. Under various geometric assumptions on X we identify some associated interpolation spaces, in particular certain real interpolation spaces. These are identified with a new scale of function spaces, which we call Z -spaces, that have recently appeared in the work of Barton and Mayboroda on elliptic boundary value problems with boundary data in Besov spaces. We also prove Hardy–Littlewood–Sobolev-type embeddings between weighted tent spaces.     

Different degrees of non-compactness for optimal Sobolev embeddings

The structure of non-compactness of optimal Sobolev embeddings of m-th order into the class of Lebesgue spaces and into that of all rearrangement-invariant function spaces is quantitatively studied. Sharp two-sided estimates of Bernstein numbers of such embeddings are obtained. It is shown that, whereas the optimal Sobolev embedding within the class of Lebesgue spaces is finitely strictly singular, the optimal Sobolev embedding in the class of all rearrangement-invariant function spaces is not even strictly singular.     

ON THE FUNCTIONAL CALCULUS IN ARCHIMEDEAN RIESZ SPACES WITH APPLICATIONS TO APPROXIMATION THEOREMS

ABSTRACT

We show that the functional calculus defined on the class of Dedekind σ-complete Riesz spaces can be extended to the class of uniformly complete Archimedean Riesz spaces without representing in the process the spaces involved by spaces of functions. As a consequence some results in the theory of Riesz spaces which were proved previously by representation techniques, can now be proved in an intrinsic way.     

On finite-to-one open mappings

We investigate open finite-to-one (mainly, k-to-one) mappings of topological spaces. The notion of a rigid space is introduced. It is shown that the class of rigid spaces contains all Euclidean spaces and all zero-dimensional Lindelöf spaces. Examples are constructed and problems are formulated.     

本性不可比的Banach空间

通过引入本性严格奇异算子和投影严格奇异算子,讨论本性不可比的Banach空间,对M.Gonzalez提出的猜想作出部分的肯定回答,最后,利用空间的本性不可比构造了一类空间,其上的非本性算子类和严格余奇异算子类是不重合的。     

The Weak Chebyshev X-Greedy Algorithm in the unweighted Bergman space

We study a greedy algorithm called the Weak Chebyshev X-Greedy Algorithm (WCXGA) and investigate its application to unweighted Bergman spaces. We first show that the WCXGA converges for a wide class of real and complex Banach spaces and dictionaries. We then prove that certain Bergman spaces and their holomorphic monomial dictionaries belong to the class of Banach spaces for which the WCXGA converges.     

Double solid twistor spaces: the case of arbitrary signature

In a recent paper ([9]) we constructed a series of new Moishezon twistor spaces which are a kind of variant of the famous LeBrun twistor spaces. In this paper we explicitly give projective models of another series of Moishezon twistor spaces on n CP ² for arbitrary n≥3, which can be regarded as a generalization of the twistor spaces of ‘double solid type’ on 3CP ² studied by Kreußler, Kurke, Poon and the author. Similarly to the twistor spaces of ‘double solid type’ on 3CP ², projective models of the present twistor spaces have a natural structure of double covering of a CP ²-bundle over CP ¹. We explicitly give a defining polynomial of the branch divisor of the double covering, whose restriction to fibers is degree four. If n≥4 these are new twistor spaces, to the best of the author’s knowledge. We also compute the dimension of the moduli space of these twistor spaces. Differently from [9], the present investigation is based on analysis of pluri-(half-)anticanonical systems of the twistor spaces.     

The law of large numbers in locally convex spaces

Summary The law of large numbers is extended to random elements taking values in locally convex spaces. The necessary and sufficient conditions for the law are given in a large class of locally convex spaces, vix. normed spaces. This class includes, among others, the test function spaces and the distribution spaces.