Structure of spaces ofC ∞-functions on nuclear spaces

LetE be a real nuclear locally convex space; we prove that the space ℰ_ub(E), of allC ^∞-functions of uniform bounded type onE, coincides with the inductive limit of the spaces ℰ_Nbc(E _v) (introduced by Nachbin-Dineen), whenV ranges over a basis of convex balanced 0-neighbourhoods inE. LetE be a real nuclear bornological vector space; we prove that the space ℰ(E) of allC ^∞-functions onE coincides with the projective limit of the spaces ℰ_Nbc(E _B), whenB is a closed convex balanced bounded subset ofE. As a consequence we obtain some density results and a version of the Paley-Wiener-Schwartz theorem. Research done during the stay of this author at the University of Bordeaux (France) in the academic year 1980–1981.     

Imbedding of power series spaces and spaces of analytic functions

The diametral dimension of a nuclear Fréchet spaceE, which satisfies (DN) and (Ω), is related to power series spaces Λ₁(ε) and Λ_∞(ε) for some exponent sequence ε. It is proved thatE contains a complemented copy of Λ_∞(ε) provided the diametral dimensions ofE and Λ_∞(ε) are equal and ε is stable. Assuming Λ₁(ε) is nuclear, any subspace of Λ₁(ε) which satisfies (DN), can be imbedded intoE. Applications of these results to spaces of analytic functions are given. Support of Turkish Scientific and Technical Research Council is gratefully acknowledged.     

On relations between non-Archimedean power series spaces

The power series spaces of finite type, A₁(α), and infinite type, A_∞(α), are the most known and important examples of non-Archimedean nuclear Fréchet spaces. We study when Aπ (α) has a subspace (or quotient) isomorphic to A_q(b).     

Metric tangent spaces to Euclidean spaces

We prove that the metric spaces pretangent to a finite-dimensional Euclidean or unitary space E are isometric to E. As a consequence of this result, we describe the metric pretangent spaces at the nonsingular points of smooth surfaces. It is also proved that there exist the spaces pretangent to the Hilbert space l ₂ , which are not isometric to it.     

On c-realcompact spaces

Abstract

The c-realcompact spaces are fully studied and most of the important and well-known properties of realcompact spaces are extended to these spaces. For a zero-dimensional space X, the space υ₀X, which is the counterpart of υX, the Hewitt realcompactification of X, is introduced and studied. It is shown that υ₀X, which is the smallest c-realcompact space between X and β₀X, plays the same role (with respect to C_c(X)) as υX does in the context of C(X). It is proved for strongly zero-dimensional spaces, c-realcompact spaces, realcompact spaces and N-compact spaces coincide. In particular, if X is a strongly zero-dimensional space, then υX = υ₀X. It is obsesrved that a zero-dimensional space X is pseudocompact if and only if C_c(X) = C*_c(X), or equivalently if and only if υ₀X = β₀ X. In particular, a zero-dimensional pseudocompact space is compact if and only if it is c-realcompact. It is shown that Lindelöf spaces, subspaces of the one-point compactification (resp., Lindelöffication) of a discrete space with a nonmeasurable cardinal, are c-realcompact space. If X is a pseudocompact space, it is observed that C(X) = C_c(X) if and only if βX is scattered. Finally, the simplest possible proof (with reasoning) among the known proofs, of the well-known fact that discrete spaces of cardinality less than or equal to that of the continuum are realcompact, is given.     

Hilbert-generated spaces

We classify several classes of the subspaces of Banach spaces X for which there is a bounded linear operator from a Hilbert space onto a dense subset in X. Dually, we provide optimal affine homeomorphisms from weak star dual unit balls onto weakly compact sets in Hilbert spaces or in c₀(Γ) spaces in their weak topology. The existence of such embeddings is characterized by the existence of certain uniformly Gâteaux smooth norms.     

On K-groups of operator algebras on HD n spaces and QD n spaces

We obtain the K-groups of the operator ideals contained in the class of Riesz operators. And based on the results, we calculate the K-groups of the operator algebras on HD nspaces and QDn spaces.     

Generalized Besov spaces and Triebel-Lizorkin spaces

In this paper the classical Besov spaces Bsp.q and Triebel-Lizorkin spaces Fsp.q for s ∈R are generalized in an isotropy way with the smoothness weights {|2j|aln}∞j=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by Bap.q and Fap.q for a ∈Irk and k ∈N, respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters a, and duality for index 0 < p < ∞. By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between Bs,p.q and ∪tsBt,p.q,and between Fsp.q and ∪ts Ftp.q, respectively. Between Bs,p,q and ∪tsBt,p.qq,and between Fsp,qand ∪tsFtp.q,respectively.     

Random processes in Sobolev-Orlicz spaces

We establish conditions under which the trajectories of random processes from Orlicz spaces of random variables belong with probability one to Sobolev-Orlicz functional spaces, in particular to the classical Sobolev spaces defined on the entire real axis. This enables us to estimate the rate of convergence of wavelet expansions of random processes from the spaces L _p (Ω) and L ₂ (Ω) in the norm of the space L _q (ℝ). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1340–1356, October, 2006.     

Baire functions and spaces of Baire functions

The paper considers (1) the tightness of spaces of Baire functions and their subspaces endowed with the topology of pointwise convergence; (2) Z _σ-mappings of K-analytic spaces; (3) K _σ-analytic spaces (Tychonoff spaces that are Z _σ-images of K-analytic spaces). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 4, pp. 3–39, 2003.     

Filter spaces

The category FIL of filter spaces and cauchy maps is a topological universe. This paper establishes the foundation for a completion theory forT ₂ filter spaces.     

New separation axioms in generalized topological spaces

We introduce and study new separation axioms in generalized topological spaces, namely, m-T_\frac14\mu\mbox{-}T_{\frac{1}{4}}, m-T_\frac38\mu \mbox{-}T_{\frac{3}{8}} and m-T_\frac12\mu\mbox{-}T_{\frac{1}{2}}. m-T_\frac14\mu\mbox{-}T_{\frac{1}{4}} spaces are strictly placed between μ-T ₀ spaces and m-T_\frac38\mu\mbox{-}T_{\frac{3}{8}}, m-T_\frac38\mu\mbox{-}T_{\frac{3}{8}} spaces are strictly placed between m-T_\frac14\mu\mbox{-}T_{\frac{1}{4}} spaces and m-T_\frac12\mu \mbox{-}T_{\frac{1}{2}} spaces, and m-T_\frac12\mu\mbox{-}T_{\frac{1}{2}} spaces are strictly placed between m-T_\frac38\mu\mbox{-}T_{\frac{3}{8}} spaces and μ-T ₁ spaces.     

A class of Banach sequence spaces analogous to the space of Popov

Hagler and the first named author introduced a class of hereditarily l ₁ Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily l _p Banach spaces for 1 ⩽ p < ∞. Here we use these spaces to introduce a new class of hereditarily l _p (c ₀) Banach spaces analogous of the space of Popov. In particular, for p = 1 the spaces are further examples of hereditarily l ₁ Banach spaces failing the Schur property.     

Extremely non-complex spaces

We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces X such that the norm equality Id+T²=1+T² holds for every bounded linear operator . This answers in the positive Question 4.11 of [V. Kadets, M. Martín, J. Merí, Norm equalities for operators on Banach spaces, Indiana Univ. Math. J. 56 (2007) 2385–2411]. More concretely, we show that this is the case of some C(K) spaces with few operators constructed in [P. Koszmider, Banach spaces of continuous functions with few operators, Math. Ann. 330 (2004) 151–183] and [G. Plebanek, A construction of a Banach space C(K) with few operators, Topology Appl. 143 (2004) 217–239]. We also construct compact spaces K₁ and K₂ such that C(K₁) and C(K₂) are extremely non-complex, C(K₁) contains a complemented copy of C(2^ω) and C(K₂) contains a (1-complemented) isometric copy of ℓ_∞.     

Uniqueness of unconditional bases in quasi-banach spaces with applications to Hardy spaces

We prove some general results on the uniqueness of unconditional bases in quasi-Banach spaces. We show in particular that certain Lorentz spaces have unique unconditional bases answering a question of Nawrocki and Ortynski. We then give applications of these results to Hardy spaces by showing the spacesH _p (T ⁿ ) are mutually non-isomorphic for differing values ofn when 0<p<1. The research of the first two authors was partially supported by NSF-grant DMS 8901636.     

Convolution operators on spaces of entire functions

We show that nontrivial convolution operators on certain spaces of entire functions on E are frequently hypercyclic when E is a normed space and when E is the strong dual of a Fréchet nuclear space. We also obtain results of existence and approximation for convolution equations on certain spaces of entire functions on arbitrary locally convex spaces.     

On banach spaces whose duals areL 1 spaces

A structure theorem for Banach spaces whose duals areL ₁ spaces, is proved. The research of the second named author has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18 through the European Office of Aerospace Research (OAR) United States Air Force.     

Nuclearity for dual operator spaces

In this short paper, we study the nuclearity for the dual operator space V* of an operator space V. We show that V* is nuclear if and only if V*** is injective, where V*** is the third dual of V. This is in striking contrast to the situation for general operator spaces. This result is used to prove that V** is nuclear if and only if V is nuclear and V** is exact.     

Uniform rotundity of Musielak-Orlicz sequence spaces

We present a criterion for uniform rotundity of Musielak-Orlicz sequence spaces. In particular, we get a better characterization of uniform rotundity of Banach spaces l({p_i}), called Nakano spaces, considered by K. Sundaresan (Studia Math. 39 (1971), 227–331.     

Finite metric spaces needing high dimension for lipschitz embeddings in banach spaces

We construct a sequence of metric spaces (M _n) with cardM _n=3n satisfying that for everyc<2, there exists a real numbera(c)>0 such that, if the Lipschitz distance fromM _n to a subset of a Banach spaceE is less thanc, then dim(E) ≥a(c)n. We also prove several results about embeddings of metric spaces whose non-zero distance values are in the interval [1,2].