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

By Pandelis Dodos

This quantity offers with difficulties within the constitution concept of separable infinite-dimensional Banach areas, with a imperative specialize in universality difficulties. This subject is going again to the beginnings of the sector and looks in Banach's classical monograph. the newness of the procedure lies within the incontrovertible fact that the solutions to a few simple questions are in keeping with suggestions from Descriptive Set conception. even supposing the ebook is orientated on proofs of numerous structural theorems, primarily textual content readers also will discover a designated exposition of diverse “intermediate” effects that are attention-grabbing of their personal correct and feature confirmed to be worthy in different components of practical research. furthermore, numerous famous leads to the geometry of Banach areas are awarded from a latest perspective.

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For every n < k let on = min{|t| : t ∈ sn,k ∩ supp(xk )} and set sn,k = {t ∈ sn,k : |t| < on }. Clearly sn,k is a segment of T with sn,k ∩ TA = ∅. The sequence (xn ) is block and so sn,k ∩ supp(xn ) ⊆ sn,k for every n < k. In fact sn,k is just the maximal initial subsegment of sn,k that does not intersect supp(xk ). Hence Psn,k (xn ) ≥ ε for every n < k. Let C = sup{ xn : n ∈ N} < +∞. 9. For every k ≥ 1 we have |{sn,k : n < k}| ≤ C 2 /ε2 . 9. Let s0 , . . , sl−1 be an enumeration of the set in question.

We proceed to give an estimate of the order of the tree TNC (Y, X, (en )). First we notice that for every Y ∈ NCX we have o TNC (Y, X, (en )) = sup o TNC (Y, X, (en ), k) : k ≥ 1 + 1. Also observe that for every k ∈ N with k ≥ 1 we have o TNC (Y, X, (en ), k) ≤ o TNC (Y, X, (en ), k) . Concerning the opposite inequality the following holds. 18. Let Y ∈ NCX . Also let C ≥ 1 and k ∈ N be such that 2C 2 ≤ k. Then o TNC (Y, X, (en ), C) ≤ o TNC (Y, X, (en ), k) . In particular, o TNC (Y, X, (en )) = sup o TNC (Y, X, (en ), δ) : δ ≥ 1 + 1.

D. Let X be a Schauder tree basis such that for every σ ∈ [T ] the sequence (xσ|n )n≥1 is unconditional. Then the basis (etn ) of T2X is unconditional. Let Y be a subspace of T2X. Assume that there exist σ ∈ [T ] and a further subspace Y of Y such that the operator Pσ : Y → Xσ is an isomorphic embedding. In such a case, the subspace Y “contains information” about the Schauder tree basis X. On the other hand, there are subspaces of T2X which are “orthogonal” to every Xσ . These subspaces are naturally distinguished into three categories, as follows.

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