There are many types of convergence that cannot be defined in terms of a norm. In this chapter we will examine the problem of the validity of Steinitz’s theorem when the convergence of series is understood in the sense of the weak topology of a normed space (§1), or when one deals with convergence in measure for series of functions (§2), and finally when one deals with convergence of series in a nuclear Fréchet space (§3). Steinitz-type problems in topological vector spaces are relatively poorly investigated, and we collected here results that are the most definitive in nature. In the first two sections we construct a number of counterexamples, and in the third section we give a result of W. Banaszczyk which extends Steinitz’s theorem from finite-dimensional spaces to nuclear Fréchet spaces.
Weitere Kapitel dieses Buchs durch Wischen aufrufen
- Rearrangements of Series in Topological Vector Spaces
Mikhail I. Kadets
Vladimir M. Kadets
- Birkhäuser Basel
- Chapter 8
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