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Series in Banach spaces: conditional and unconditional convergence. Transl. from the Russian by Andrei Iacob. (English) Zbl 0876.46009

Operator Theory: Advances and Applications. 94. Basel: Birkhäuser. viii, 156 p. (1997).
There are two fundamental problems concerning series \((*)\) \(\sum x_k\), where \(x_k\in X\):
1) Characterize unconditionally convergent series \((*)\),
2) If the series \((*)\) is conditionally convergent, determine its sum range \(SR(*)\), i.e. the set of sums of all convergent rearrangements.
In case \(X=\mathbb{R}\), unconditional convergence of \((*)\) is equivalent to its absolute convergence and, by Riemann’s theorem \(SR(*)= \overline{R}= [-\infty,\infty]\).
The main topic of the book are results concerning the second problem in case when \(X=\mathbb{R}^n\), \(X\) is a Banach space and \(X\) is a topological vector space (t.v.s.). After preliminary material in Chapter 1, there is considered in Chapter 2 the case \(X=\mathbb{R}^n\). The starting point is Steinitz’s theorem (1913) stating that \(SR(*)\) is an affine space which does not need to coincide with \(\mathbb{R}^n\). D. V. Perchnskii’s theorem and S. A. Chobanyan’s inequalities end Chapter 2. Conditional convergence in infinite-dimensional spaces is considered in Chapter 3. It contains basic counterexamples to Steinitz theorem in this case, showing among else that \(SR(*)\) may consist of one point or of two points. Chobanian’s theorem is obtained applying probabilistic inequalities, and M. I. Kadets’ theorem is derived. Chapter 4 is devoted to unconditionally convergent series; there are presented the Dvoretzky-Rogers theorem and the famous Orlicz theorem for \(X=L^p\). This topic is followed also in Chapter 5, where the Banach-Mazur distance \(d(X,Y)\) between Banach spaces \(X\) and \(Y\) is applied. Chapter 6 contains material needed in the sequel, especially Dvoretzky’s theorem on almost Euclidean sections and Mazur’s theorem an basic sequences. In Chapter 7 the authors turn back to problems of Steinitz’s theorem in case of infinite-dimensional Banach spaces \(X\). Here, the infratype \(p\) of \(X\) is of importance and results by V. M. Kadets are referred. Among else it follows that in any infinite-dimensional Banach space there are series \((*)\) such that \(SR(*)\) consists exactly of two points. Results when \(X\) is a t.v.s. are presented in Chapter 8. The valadity of the Steinitz theorem is considered in cases when convergence is understood in the weak topology of a normed space or in the sense in measure for series of functions. Recent results of W. Banaszczyk concerning the validity of Steinitz’s theorem in nuclear spaces \(X\) are presented. In an Appendix there are referred recent results on sets \(I(f)\) of Riemann integral sums for bounded vector valued functions \(f\).
The material in the book is always started with motivation. There are plenty of exercises, with hints for solution. The list of references is complete and up-to-day. This is an amazing book on doctoral level, written by experienced authors, who contributed a lot to the subject presented in the book.

MSC:

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B20 Geometry and structure of normed linear spaces
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46A35 Summability and bases in topological vector spaces
46B09 Probabilistic methods in Banach space theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26E20 Calculus of functions taking values in infinite-dimensional spaces
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
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