Probability and Measure

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Educational Assessment Of Students

It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, that is, that any analytic equivalence relation Borel reduces to it. This solves a problem of G. Godefroy. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for: (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomorphism, and (5) Schauder bases up to permutative equivalence. Some of the constructions rely on methods recently developed by S. Argyros and P. Dodos.

/pdf/the-complexity-of-classifying-separable-banach-spaces-up-to-5ez5a4vlpp.pdf

The complexity of classifying separable Banach spaces up to isomorphism

Review: J. D. Halpern, H. Lauchli, A Partition Theorem; J. D. Halpern, A. Levy, The Boolean Prime Ideal Theorem Does Not Imply the Axiom of Choice

Consequences of Martin's axiom: When m K > ω1

Genericity and amalgamation of classes of Banach spaces

Basic Concepts- The Standard Borel Space of All Separable Banach Spaces- The ?2 Baire Sum- Amalgamated Spaces- Zippin's Embedding Theorem- The Bourgain-Pisier Construction- Strongly Bounded Classes of Banach Spaces

/pdf/banach-spaces-and-descriptive-set-theory-selected-topics-27u0rmgnw6.pdf

Banach Spaces and Descriptive Set Theory: Selected Topics

It is shown that for every separable Banach space X with non-separable dual, the space $X^{**}$ contains an unconditional family of size $|X^{**}|$ . The proof is based on Ramsey Theory for trees and finite products of perfect sets of reals. Among its consequences, it is proved that every dual Banach space has a separable quotient.

/pdf/unconditional-families-in-banach-spaces-3308jmz72q.pdf

Unconditional families in Banach spaces

We show that the classes of separable reflexive Banach spaces and of spaces with separable dual are strongly bounded. This gives a new proof of a recent result of E. Odell and Th. Schlumprecht, asserting that there exists a separable reflexive Banach space containing isomorphic copies of every separable uniformly convex Banach spaces.

/pdf/some-strongly-bounded-classes-of-banach-spaces-2mttdwfgv3.pdf

Some strongly bounded classes of Banach spaces

V. D. Milman proved in [20] that the product of two strictly singular operators on L
 p
 [0, 1] (1 ⩽ p < 1) or on C[0, 1] is compact. In this note we utilize Schreier families $$
\mathcal{S}_\xi 
$$
 in order to define the class of $$
\mathcal{S}_\xi 
$$
 -strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying Banach space has finitely many equivalence classes of Schreier-spreading sequences. Finally we define the class of $$
\mathcal{S}_\xi 
$$
 -hereditarily indecomposable Banach spaces and we examine the operators on them.

/pdf/classes-of-strictly-singular-operators-and-their-products-humirwovck.pdf

Pandelis Dodos

Papers

Genericity and amalgamation of classes of Banach spaces

Banach Spaces and Descriptive Set Theory: Selected Topics

Unconditional families in Banach spaces

Some strongly bounded classes of Banach spaces

Classes of strictly singular operators and their products