By Pandelis Dodos

ISBN-10: 3642121527

ISBN-13: 9783642121524

This quantity offers with difficulties within the constitution idea of separable infinite-dimensional Banach areas, with a primary specialise in universality difficulties. This subject is going again to the beginnings of the sphere and looks in Banach's classical monograph. the newness of the technique lies within the undeniable fact that the solutions to a few easy questions are in response to suggestions from Descriptive Set thought. even if the ebook is orientated on proofs of a number of structural theorems, in general textual content readers also will discover a certain exposition of various “intermediate” effects that are attention-grabbing of their personal correct and feature confirmed to be priceless in different components of practical research. in addition, numerous recognized ends up in the geometry of Banach areas are offered from a latest perspective.

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**Example text**

13. 13. Let A ⊆ S × H be as in the statement of the proposition. 14, the map ΦA : S → K(H) deﬁned by ΦA (s) = As is Borel, and so, the set {As : s ∈ S} is an analytic subset of K(H). Let n ∈ N. 10, the set ΩDn = {K ∈ K(H) : Dn∞ (K) = ∅} is Π11 and the map K → |K|Dn is a Π11 -rank on ΩDn . 9, we see that for every s ∈ S and for every ξ < ω1 if Dnξ (As ) = ∅, then Dnξ+1 (As ) Dnξ (As ). It follows that the sequence {Dnξ (As ) : ξ < ω1 } of iterated derivatives of As must be stabilized at ∅. Hence {As : s ∈ S} ⊆ ΩDn .

Then the following are satisﬁed: (i) If Y is X-compact, then Y is X-singular. (ii) If Y is X-singular, then Y is weakly X-singular. Proof. Part (i) is straightforward. To see part (ii) let Y be an X-singular subspace of T2X and let A be an arbitrary ﬁnite subset of [T ]. Consider the B-tree TA generated by A. Notice that there exist ﬁnal segments s0 , . . , sk of T and a ﬁnite-dimensional subspace F of T2X such that XTA ∼ = F ⊕ k ⊕X . 6, if the operator P : Y → X were an sn TA TA n=0 40 3 The 2 Baire Sum isomorphic embedding, then there would existed some σ ∈ A and a subspace Y of Y such that the operator Pσ : Y → Xσ is an isomorphic embedding too.

Then for every ε > 0 there exists a normalized block sequence (yn ) is Y such that lim sup Pσ (yn ) < ε for every σ ∈ [T ]. Proof. The proof is a quest of a contradiction. So, suppose that there exist a block weakly X-singular subspace Y of T2X and ε > 0 such that for every normalized block sequence (yn ) in Y there exists σ ∈ [T ] such that lim sup Pσ (yn ) ≥ ε. Let p ∈ N and r > 0 to be determined later. We start with a normalized block sequence (yn0 ) in Y . By our assumptions, there exist σ0 ∈ [T ] and L0 ∈ [N] such that Pσ0 (yn0 ) > ε/2 for every n ∈ L0 .

### Banach Spaces and Descriptive Set Theory: Selected Topics by Pandelis Dodos

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