Resolvability of Lindelöf spaces

We prove the ω-resolvability of hereditarily finally compact spaces and the resolvability of Lindelöf spaces whose dispersion character is uncountable.     

Remarks on discretely absolutely star-Lindelöf spaces

In this paper, we prove the following statements(1) There exists a Hausdorff Lindelöf space X such that the Alexandroff duplicate A(X) of X is not discretely absolutely star-Lindelöf.(2) If X is a regular Lindelöf space, then A(X) is discretely absolutely star-Lindelöf.(3) If X is a normal discretely star-Lindelöf space with e(X) < ω ₁, then A(X) is discretely absolutely star-Lindelöf.     

Some generalizations of Lindelöf and paracompact spaces

In this paper notions ofm-Lindelöf, meta-m-Lindelöf, para-m-Lindelöf andm-closure preserving property are defined, wherem is any infinite cardinal. The main results are the following:

1. A topological space ism-Lindelöf if and only if it is meta-m-Lindelöf and it ism-Lindelöf in the sense of complete accumulation point.
2. A regular topological space is paracompact if and only if it is para-m-Lindelöf and it hasm-closure preserving property for somem.

     

Some remarks on generalizations of countably compact spaces and Lindelöf spaces

We prove some propositions and present some examples concerning the properties between countably compact and pseudocompact and the properties between Lindelöf and pseudo-Lindelöf.     

Monotone weak Lindelöfness

The definition of monotone weak Lindelöfness is similar to monotone versions of other covering properties: X is monotonically weakly Lindelöf if there is an operator r that assigns to every open cover U a family of open sets r(U) so that (1) ∪r(U) is dense in X, (2) r(U) refines U, and (3) r(U) refines r(V) whenever U refines V. Some examples and counterexamples of monotonically weakly Lindelöf spaces are given and some basic properties such as the behavior with respect to products and subspaces are discussed.     

A result on monotonically Lindelöf generalized ordered spaces

In this paper, we show that a generalized ordered space representable as the union of two closed monotonically Lindelöf subspaces is monotonically Lindelöf, which partially answers a question [7, Question 2] of Levy and Matveev. In addition, we show that the monotone Lindelöf property is hereditary with respect to open Lindelöf subsets in generalized ordered spaces.     

Monotone operators on Gödel logic

We consider an extension of Gödel logic by a unary operator that enables the addition of non-negative reals to truth-values. Although its propositional fragment has a simple proof system, first-order validity is Π ₂-hard. We explain the close connection to Scarpellini’s result on Π ₂-hardness of ?ukasiewicz’s logic.     

On countable unions of nonmeager sets in hereditarily Lindelöf spaces

It is well known that any Vitali set on the real line ? does not possess the Baire property. The same is valid for finite unions of Vitali sets. What can be said about infinite unions of Vitali sets? Let S be a Vitali set, S _r be the image of S under the translation of ? by a rational number r and F = {S _r : r is rational}. We prove that for each non-empty proper subfamily F′ of F the union ∪F′ does not possess the Baire property. We say that a subset A of ? possesses Vitali property if there exist a non-empty open set O and a meager set M such that A ? O \ M. Then we characterize those non-empty proper subfamilies F′ of F which unions ∪F′ possess the Vitali property.     

Notes on pre-Frölicher spaces

In this work we introduce a class of Sikorski differential spaces (M, D) called pre-Fr¨olicher spaces, on which the process of yielding a Fr¨olicher structure on the underlying set M is D preserving, their category we denote by preFrl. We investigate some algebraic properties on these spaces whose subsequent geometric properties are mostly similar to those of smooth manifolds, except for the invariance of dimension, and also that preFrl naturally induces a Cartesian closed subcategory of the category Frl in which there is no discrete object. Using this Cartesian property, it is shown that the Gelfand representation is a smooth map, that the tangent as well as cotangent bundles are made smooth spaces in an unusual but more natural way via smooth curves.     

Tensor product representation of Köthe-Bochner spaces and their dual spaces

We provide a tensor product representation of Köthe-Bochner function spaces of vector valued integrable functions. As an application, we show that the dual space of a Köthe-Bochner function space can be understood as a space of operators satisfying a certain extension property. We apply our results in order to give an alternate representation of the dual of the Bochner spaces of p-integrable functions and to analyze some properties of the natural norms \(\Delta _p\) that are defined on the associated tensor products.     

Monotone Iterationsfolgen und ihre Verwendung zur Lösung linearer Gleichungssysteme

Ohne ZusammenfassungAuszug aus der von der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Hamburg angenommenen Habilitationsschrift des Verfassers (Referenten: Prof. Dr. Dr. h. c.L. Collatz und Prof. Dr.E. Sperner); ein anderer Auszug ist bereits unter dem Titel Fehlerabschätzungen bei Relaxationsverfahren zur numerischen Auflösung linearer Gleichungssysteme erschienen [Num. Math.3, 188–201 (1961)].     

Hölder type inequalities in Lorentz spaces

We study optimal Hölder type inequalities for the Lorentz spaces L ^p,s (R, μ), in the range 1 < p < ∞, 1 ≤ s ≤ ∞, for both the maximal and the dual norms. These estimates also give sharp results for the corresponding associate norms.     

On elliptic systems in Hörmander spaces

We study a linear system of pseudodifferential equations uniformly elliptic in Petrovskii’s sense in the Hilbert scale of H?rmander functional spaces defined in ℝ _n . An a priori estimate is proved for the solution of the system and its interior smoothness in this scale of spaces is investigated. As an application, we establish a sufficient condition for the existence of continuous bounded derivatives of the solution.     

Holomorphic Hölder-type spaces and composition operators

We study boundedness and compactness of composition operators in the generalized Hölder-type space of holomorphic functions in the unit disc with prescribed modulus of continuity. We also devote a significant part of the article to outline some embeddings between such Hölder-type spaces, to discuss properties of modulus of continuity and to construct some useful examples.     

Diagonal multilinear operators on Köthe sequence spaces

We analyse the interplay between maximal/minimal/adjoint ideals of multilinear operators (between sequence spaces) and their associated Köthe sequence spaces. We establish relationships with spaces of multipliers and apply these results to describe diagonal multilinear operators from Lorentz sequence spaces. We also define and study some properties of the ideal of (E, p)-summing multilinear mappings, a natural extension of the linear ideal of absolutely (E, p)-summing operators.     

On quasimöbius maps in real Banach spaces

Suppose that E and E′ denote real Banach spaces with dimension at least 2, that D $ \subseteq $ E and D′ $ \subseteq $ E′ are domains, that f: D → D′ is an (M,C)-CQH homeomorphism, and that D is uniform. The aim of this paper is to prove that D′ is a uniform domain if and only if f extends to a homeomorphism $\overline f :\overline D \to {\overline D ^\prime }$ and $\overline f $ is η-QM relative to ?D. This result shows that the answer to one of the open problems raised by Väisälä from 1991 is affirmative.     

Degenerate differential operators in weighted Hölder spaces

A differential operator ?, arising from the differential expression $$lv(t) \equiv ( - 1)^r v^{[n]} (t) + \sum\nolimits_{k = 0}^{n - 1} {p_k } (t)v^{[k]} (t) + Av(t),0 \leqslant t \leqslant 1,$$ , and system of boundary value conditions $$P_v [v] = \sum\nolimits_{k = 0}^{n_v } {\alpha _{vk} } r^{[k]} (1) = 0.v - 1, \ldots ,\mu ,0 \leqslant \mu< n$$ is considered in a Banach space E. Herev ^[k](t)=(a(t) d/dt) ^k v(t)a(t) being continuous fort?0, α(t) >0 for t > 0 and \(\int_0^1 {\frac{{dz}}{{a(z)}} = + \infty ;}\) the operator A is strongly positive in E. The estimates , are obtained for ?: n even, λ varying over a half plane.