Partial inner product spaces and semi-inner product spaces

A comparison is made between the two objects mentioned in the title. Connections between them are threefold: (i) both are particular instances of dual pairs of locally convex spaces; (ii) many partial inner product spaces consist of chains or lattices of semi-inner product spaces; (iii) the basic structure behind both of them is that of Galois connections. A number of common open problems are described.     

The approximate fixed point property in product spaces

In this paper we generalize to unbounded   convex subsets C$C$ of hyperbolic   spaces results obtained by W.A. Kirk and R. Espínola on approximate fixed points of nonexpansive mappings in product spaces _(C×M)∞${(C\times M)}_{\infty }$, where M$M$ is a metric space and C$C$ is a nonempty, convex, closed and bounded subset of a normed or a CAT(0)-space. We extend the results further, to families _(Cu)u∈M${\left(C_u\right)}_{u\in M}$ of unbounded convex subsets of a hyperbolic space. The key ingredient in obtaining these generalizations is a uniform quantitative version of a theorem due to Borwein, Reich and Shafrir, obtained by the authors in a previous paper using techniques from mathematical logic. Inspired by that, we introduce in the last section the notion of uniform approximate fixed point property   for sets C$C$ and classes of self-mappings of C$C$. The paper ends with an open problem.     

Sequential closure in product spaces

Let X denote the product of m-many second countable Hausdorff spaces. Main theorems: (1) If S?X is invariant under compositions, m is weakly accessible (resp., nonmeasurable), and F?S is sequentially closed and a sequential G_σ-set which is invariant under projections for finite sets (resp., F?S is sequentially open and sequentially closed), then F is closed. (2) If S?X is invariant under projections and m is nonmeasurable, then every sequentially continuous {0, 1} valued function on S is continuous. (3) A sequentially continuous {0, 1}-valued function on an m-adic space of nonmeasurable weight is continuous. Now let X denote the product of arbitrarily many W-spaces and S?X be invariant under compositions. (4) Then in S, the closure of any Q-open subset coincides with its sequential closure.     

K-spaces property of product spaces

LetK be a class of spaces which are eigher a pseudo-opens-image of a metric space or ak-space having a compact-countable closedk-network. LetK′ be a class of spaces which are either a Fréchet space with a point-countablek-network or a point-G _δ k-space having a compact-countablek-network. In this paper, we obtain some sufficient and necessary conditions that the products of finitely or countably many spaces in the classK orK′ are ak-space. The main results are that

The multiplier operators on the weighted product spaces

In this paper, we proved the boundedness of multiplier operators on the weighted product spaces.

     

The Mazur product map on Hardy-type sequence spaces

We study certain Hardy-type sequence spaces Hp and , 1?p?∞, which are analogues of ?^∞ and c₀, respectively. We show that the Mazur product is not onto for every p∈(1,∞) with q=p⁻¹(p−1). We present corollaries for spaces defined via weighted ?p seminorms and for c₀. The latter corollary provides a new solution of Mazur's Problem 8 in the Scottish Book.     

The topological product structure of systems of Lebesgue spaces

The first author was in part supported by the Research Grants Committee of the University of Alabama-Project 1530     

On shape of product spaces

It is known that if X is a compactum and Y is metrizable Sh₅(X × Y) is not generally determined by Sh₅(X) and Sh₅(Y), where Sh₅(Z) is the strong shape of Z in the sense of Borsuk. In this paper it is proved that Sh(X × Y) is uniquely determined by Sh(X) and Sh(Y), where Sh(Z) is the shape of Z in the sense of Fox. If X is an FANR and Y is an MANR, then X × Y is an MANR.     

Anisotropic singular integrals in product spaces

In this paper, the authors introduce a class of product anisotropic singular integral operators, whose kernels are adapted to the action of a pair A := (A1, A2) of expansive dilations on R n and R m , respectively. This class is a generalization of product singular integrals with convolution kernels introduced in the isotropic setting by Fefferman and Stein. The authors establish the boundedness of these operators in weighted Lebesgue and Hardy spaces with weights in product A∞ Muckenhoupt weights on R n × R m . These results are new even in the unweighted setting for product anisotropic Hardy spaces.     

New concentration inequalities in product spaces

We introduce three new ways to measure the “distance” from a point to a subset of a product space and we prove corresponding concentration inequalities. Each of them allows to control the fluctuation of a new class of random variables. Oblatum 29-IX-1995 & 15-IV-1996     

Function spaces and product topologies on powers of spaces

A study is made of two classes of product topologies on powers of spaces: the general box product topologies, and the general uniform product topologies. Some examples are given and some results are shown about the properties of these general product spaces. This is applied to show that certain spaces of continuous functions with the fine topology are homeomorphic to box product spaces, and certain spaces of continuous functions with the uniform topology are homeomorphic to uniform product spaces.     

The dimension problem for the product of infinite-dimensional spaces

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i ee Prilozheniya. Tematicheskiye Obzory. Vol. 14, Topologiya-2, 1994