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Ideal spaces. (English) Zbl 0896.46018
Lecture Notes in Mathematics. 1664. Berlin: Springer. 146 p. DM 36.00; öS 262.80; sFr. 32.50; £15.00; $ 27.00 (1997).
Ideal spaces are normed spaces of measurable functions. They form a rather wide class which contains the Lebesgue, Lorentz, Orlicz and Marcinkiewicz spaces including their weighted and other forms. A certain problem with these spaces in existing literature is that they have different names (namely, Banach function spaces and Köthe spaces), and are defined in different ways.
The author extends the basic summarizing texts [A. C. Zaanen, “Integration”, Amsterdam (1967; Zbl 0175.05002) and P. P. Zabrejko, Ideal spaces of functions I (in Russian), Vestnik Jarosl. Univ., 12-52 (1974)] and concentrates on the general case of vector-valued functions (in particular, of functions with values in infinite-dimensional Banach spaces) on arbitrary measure spaces (not necessarily finite or \(\sigma\)-finite measure spaces), on spaces with mixed norms, and on calculus with functions with values in ideal spaces. He shows that the presented results can be applied in the theory of integro-differential equations (in particular, of Barbashin type) and that they yield in strong theorems on continuity of operators of Hammerstein type. As he admitted, the former was the main motivation for the book. The remarkable feature is that the proofs are constructive in the sense that the author avoids the use of the axiom of choice and of the continuum hypothesis.
The book is organized in five chapters (1. Introduction, 2. Basic definitions, 3. Ideal spaces with additional properties, 4. Ideal spaces on product measures and calculus, 5. Operators and applications) and one appendix (Some measurability results).

MSC:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46E40 Spaces of vector- and operator-valued functions
46A45 Sequence spaces (including Köthe sequence spaces)
46B45 Banach sequence spaces
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
28A35 Measures and integrals in product spaces
28E15 Other connections with logic and set theory
46B10 Duality and reflexivity in normed linear and Banach spaces
47B38 Linear operators on function spaces (general)
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
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