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Uniqueness of structure in Banach spaces. (English) Zbl 1058.46006
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1635-1669 (2003).
In this article, the author presents a detailed overview of the study of uniqueness of various types of bases and function structure of Banach spaces. The paper includes the following sections: 1. Uniqueness of general and unconditional bases; 2. Uniqueness of symmetric bases; 3. Uniqueness of unconditional bases, up to a permutation; 4. Uniqueness in finite-dimensional spaces; 5. Uniqueness of rearrangement invariant structures; 6. Uniqueness of bases in non-Banach spaces. The paper includes an extensive list of references. It is very well written, by one of the major contributors to the subject, and can serve both as an excellent introduction to the field and as a reference on all important developments up to date.
For the entire collection see [Zbl 1013.46001].

##### MSC:
 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B20 Geometry and structure of normed linear spaces 46B03 Isomorphic theory (including renorming) of Banach spaces 46B45 Banach sequence spaces 46B42 Banach lattices 46-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis
##### Keywords:
uniqueness of bases; unconditional bases; symmetric bases