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Banach Spaces and Descriptive Set Theory: Selected Topics 2010 ed. [Mīkstie vāki]

  • Formāts: Paperback / softback, 168 pages, height x width: 235x155 mm, weight: 580 g, XII, 168 p., 1 Paperback / softback
  • Sērija : Lecture Notes in Mathematics 1993
  • Izdošanas datums: 11-May-2010
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3642121527
  • ISBN-13: 9783642121524
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Banach Spaces and Descriptive Set Theory: Selected Topics 2010 ed.Palielināt
  • Formāts: Paperback / softback, 168 pages, height x width: 235x155 mm, weight: 580 g, XII, 168 p., 1 Paperback / softback
  • Sērija : Lecture Notes in Mathematics 1993
  • Izdošanas datums: 11-May-2010
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3642121527
  • ISBN-13: 9783642121524
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783642121524.html
Keywords:
These notes are devoted to the study of some classical problems in the Geometry of Banach spaces. The novelty lies in the fact that their solution relies heavily on techniques coming from Descriptive Set Theory. Thecentralthemeisuniversalityproblems.Inparticular,thetextprovides an exposition of the methods developed recently in order to treat questions of the following type: (Q) LetC be a class of separable Banach spaces such that every space X in the classC has a certain property, say property (P). When can we ?nd a separable Banach space Y which has property (P) and contains an isomorphic copy of every member ofC? We will consider quite classical properties of Banach spaces, such as - ing re exive, having separable dual, not containing an isomorphic copy of c , being non-universal, etc. 0 It turns out that a positive answer to problem (Q), for any of the above mentioned properties, is possible if (and essentially only if) the classC is simple. The simplicity ofC is measured in set theoretic terms. Precisely, if the classC is analytic in a natural coding of separable Banach spaces, then we can indeed ?nd a separable space Y which is universal for the class C and satis es the requirements imposed above.

Recenzijas

From the reviews:

The book under review is mainly focused on the authors work on characterizations of the existence of universal elements in subclasses of separable Banach spaces . The book collects the most important of these results into a self-contained framework that clarifies the ideas under the successive improvements between each paper and the following ones. All this makes the book a mandatory reference for anyone interested in universality in Banach spaces. (Matias Raja, Mathematical Reviews, Issue 2011 j)

The author uses descriptive set theory to prove results on the structure of Banach spaces. this book may be useful for people interested in Banach space theory or/and descriptive set theory. It is very well written and contains a lot of results and techniques from these two theories, and thus may serve as a reference book. (Daniel Li, Zentralblatt MATH, Vol. 1215, 2011)

1 Basic Concepts
1(8)
1.1 Polish Spaces and Standard Borel Spaces
1(2)
1.2 Trees
3(3)
1.3 Universal Spaces
6(2)
1.4 Comments and Remarks
8(1)
2 The Standard Borel Space of all Separable Banach Spaces
9(28)
2.1 Definitions and Basic Properties
9(4)
2.1.1 Properties of SB
10(1)
2.1.2 Coding the Dual of an X ε SB
11(2)
2.2 The Class REFL
13(3)
2.3 The Class SD
16(7)
2.3.1 The Szlenk Index
16(2)
2.3.2 Norm-Separable Compact Subsets of (Bz, w)
18(1)
2.3.3 The Szlenk Index is a II1/1-Rank on SD
19(1)
2.3.4 The Dual Class of an Analytic Subset of SD
20(3)
2.4 The Class NCx
23(2)
2.5 Coding Basic Sequences
25(5)
2.5.1 The Convergence Rank γ
26(1)
2.5.2 Subsequences Spanning Complemented Subspaces
27(2)
2.5.3 Proof of Theorem 2.20
29(1)
2.6 Applications
30(4)
2.7 Comments and Remarks
34(3)
3 The lz Baire Sum
37(20)
3.1 Schauder Tree Bases
37(1)
3.2 The l2 Baire Sum of a Schauder Tree Basis
38(2)
3.3 Weakly Null Sequences in T2χ
40(6)
3.3.1 General Lemmas
40(3)
3.3.2 Sequences Satisfying an Upper l2 Estimate
43(2)
3.3.3 Proof of Theorem 3.6
45(1)
3.4 Weakly X-Singular Subspaces
46(4)
3.5 X-Singular Subspaces
50(4)
3.6 Schauder Tree Bases Not Containing l1
54(2)
3.7 Comments and Remarks
56(1)
4 Amalgamated Spaces
57(14)
4.1 Definitions and Basic Properties
57(3)
4.2 Finding Incomparable Sets of Nodes
60(5)
4.3 Proof of Theorem 4.6
65(4)
4.4 Comments and Remarks
69(2)
5 Zippin's Embedding Theorem
71(18)
5.1 Fragmentation, Slicing, and Selection
72(5)
5.1.1 Fragmentation
72(1)
5.1.2 Slicing Associated to a Fragmentation
73(1)
5.1.3 Derivative Associated to a Fragmentation
73(2)
5.1.4 The "Last Bite" of a Slicing
75(1)
5.1.5 The "Dessert" Selection of a Fragmentation
76(1)
5.2 Parameterized Fragmentation
77(3)
5.3 The Embedding
80(4)
5.4 Parameterizing Zippin's Theorem
84(3)
5.5 Comments and Remarks
87(2)
6 The Bourgain---Pisier Construction
89(26)
6.1 Kisliakov's Extension
89(5)
6.1.1 Basic Properties
90(1)
6.1.2 Preservation of Isomorphic Embeddings
91(1)
6.1.3 Minimality
92(1)
6.1.4 Uniqueness
93(1)
6.2 Admissible Embeddings
94(5)
6.2.1 Stability Under Compositions
96(1)
6.2.2 Stability Under Quotients
96(1)
6.2.3 Metric Properties
97(2)
6.3 Inductive Limits of Finite-Dimensional Spaces
99(3)
6.4 The Construction
102(1)
6.5 Parameterizing the Construction
103(6)
6.6 Consequences
109(4)
6.6.1 A Result on Quotient Spaces
109(3)
6.6.2 Applications
112(1)
6.7 Comments and Remarks
113(2)
7 Strongly Bounded Classes of Banach Spaces
115(12)
7.1 Analytic Classes of Separable Banach Spaces and Schauder Tree Bases
115(3)
7.2 Reflexive Spaces
118(1)
7.3 Spaces with Separable Dual
119(2)
7.4 Non-universal Spaces
121(2)
7.5 Spaces Not Containing a Minimal Space X
123(2)
7.6 Comments and Remarks
125(2)
A Rank Theory
127(10)
B Banach Space Theory
137(12)
B.1 Schauder Bases
137(1)
B.2 Operators on Banach Spaces
138(1)
B.3 Interpolation Method
139(2)
B.4 Local Theory of Infinite-Dimensional Banach Spaces
141(1)
B.5 Theorem 6.13: The Radon-Nikodym Property
142(7)
C The Kuratowski-Tarski Algorithm
149(2)
D Open Problems
151(4)
References 155(4)
Index 159