Decomposition of Banach Space into a Direct Sum of Separable and Reflexive Subspaces and Borel Maps

* This paper was supported in part by the Bulgarian Ministry of Education, Science and Technologies under contract MM-506/95.The main results of the paper are: Theorem 1. Let a Banach space E be decomposed into a direct sum of separable and reflexive subspaces. Then for every Hausdorff locally convex topological vector space Z and for every linear continuous bijective operator T : E → Z, the inverse T^(−1) is a Borel map. Theorem 2. Let us assume the continuum hypothesis. If a Banach space E cannot be decomposed into a direct sum of separable and reflexive subspaces, then there exists a normed space Z and a linear continuous bijective operator T : E → Z such that T^(−1) is not a Borel map

Norm approximation property

We introduce and study a general approximation property which takes origin in Numerical Analysis

Banach spaces in various positions

AbstractWe formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index a(Y,X) that measures how many non-equivalent positions Y admits in X, and obtain estimates of a(Y,X) for X a classical Banach space such as ℓp,Lp,L1,C(ωω) or C[0,1]. Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from c0 or ℓ2? Recall that a Banach space X is said to be automorphic if every subspace Y admits only one position in X; i.e., a(Y,X)=1 for every subspace Y of X. We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an L∞-space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both X and X∗ are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either c0 or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either L∞ or a superreflexive type 2 Banach lattice

Rate of Decay of the Bernstein Numbers

We show that if a Banach space X contains uniformly complemented l₂ⁿ 's then there exists a universal constant b = b(X) > 0 such that for each Banach space Y, and any sequence dn ↓ 0 there is a bounded linear operator T : X → Y with the Bernstein numbers bn(T) of T satisfying b⁻¹dn ≤ bn(T) ≤ bdn for all n.Показано, что для B-выпуклого сепарабельного пространства X, произвольного банахова пространства Y и любой последовательности dn ↓ 0 существует такой ограниченный линейный оператор T : X → Y и b > 0, что для всех чисел Бернштейна bn(T) оператора T имеем для любого n b⁻¹dn ≤ bn(T) ≤ bdn

Banach spaces with projectional skeletons

A projectional skeleton in a Banach space is a sigma-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Sigma-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of known results concerning weakly compactly generated spaces and Plichko spaces. We prove a preservation result for Plichko Banach spaces, involving transfinite sequences of projections. As a corollary, we show that a Banach space is Plichko if and only if it has a commutative projectional skeleton.Comment: 32 pages (including index and toc); revised (added example, comments, references

Springs as indicator of geotectonic disturbances

The main object of research in this publication is the springs considered in the development system of the regional structural geology of the Carpathians. In specific theoretical and practical examples it proved that the springs can be indicators of linear faults. The effectiveness of combining field methods with remote methods for detecting geological explosive disturbances and fracturing of the local substrate has been proved. The prospects and directions for further research are shown.The main object of research in this publication is the springs considered in the development system of the regional structural geology of the Carpathians. In specific theoretical and practical examples it proved that the springs can be indicators of linear faults. The effectiveness of combining field methods with remote methods for detecting geological explosive disturbances and fracturing of the local substrate has been proved. The prospects and directions for further research are shown

МОДЕРНІЗАЦІЯ КОМПЛЕКСУ ТИПУ ІК-1У ДЛЯ ВИМІРЮВАННЯ ОПОРУ ЗАЗЕМЛЮВАЛЬНОГО ПРИСТРОЮ БЛИСКАВКОВІДВОДІВ ТА ОПОР ЛЕП

Purpose. The creation of a measuring device for determining the impedance of the grounding of lightning arresters and supports of overhead lines under the influence of aperiodic pulses with the parameters 1.2/50 μs, 8/20 μs and 10/350 μs. Methodology. For this purpose, electrical engineering theory, transient modeling software and natural modeling methods are used. Results. The parameters of the electrical circuits of the additional forming unit were determined to create lightning current pulses with parameters of 10/350 μs using the IK-1U measuring complex. According to the simulation results, a layout of the forming unit in the form of an attachment and the IK-1U complex with the upgraded power supply system were created. Oscillograms of the front and pulse duration are obtained. The specified model was tested when performing electromagnetic diagnostics of the state of the RFP for more than 100 operating electrical substations. Originality. The measuring complex IK-1U was improved, which made it possible to determine the impulse impedance of the grounding device of lightning arresters when exposed to a current of 10/350 μs, 8/20 μs and of voltages 1.2/50 μs. Practical value. Upgraded device allows measurements in accordance with modern international requirements. Метою роботи є створення вимірювального приладу для визначення опору заземлювальних пристроїв (ЗП) блискавковідводів та опор повітряних ліній електропередачі (ЛЕП) при дії аперіодичних імпульсів напруги з параметрами 1,2/50 мкс та струму з параметрами 8/20 мкс і 10/350 мкс. Для цього використано теорію електротехніки, програмні засоби моделювання перехідних процесів та методи натурного моделювання. Було визначено параметри елементів електричного кола додаткового формуючого блоку для створення грозових імпульсів струму з параметрами 10/350 мкс за допомогою вимірювального комплексу типу ІК-1У. Вдосконалено комплекс типу ІК-1У, що дозволило визначати імпульсний опір ЗП блискавковідводів та опор ЛЕП при дії імпульсів струму 10/350 мкс, 8/20 мкс та напруги 1,2/50 мкс. Модернізований прилад дозволяє проводити вимірювання відповідно з сучасними міжнародними вимогами