# 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 relevant 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 indisputable fact that the solutions to a few uncomplicated questions are in line with thoughts from Descriptive Set thought. even supposing the ebook is orientated on proofs of numerous structural theorems, more often than not textual content readers also will discover a targeted exposition of various “intermediate” effects that are fascinating of their personal correct and feature confirmed to be invaluable in different components of sensible research. additionally, numerous famous ends up in the geometry of Banach areas are provided from a latest perspective.

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Extra info for Banach Spaces and Descriptive Set Theory: Selected Topics

Example text

4 The Class NCX 23 is Borel. To see this notice that m k (yn ) ∼ (hn ) ⇔ ∀m ∀a0 , . . , am ∈ Q ∀l m an hn (l) ≤ k n=0 and ∀p ∃i 1 k m an y n − n=0 1 ≤ p+1 an y n n=0 m an hn (i) . n=0 The sequence (fn ) consists of Borel functions. Therefore, the relation Ik in C(2N )N × S deﬁned by (yn ), X ∈ Ik ⇔ (yn ), (fn (X)) ∈ Ek is Borel. Finally, by property (P5) in Sect. 1, the relation R in SB×C(2N )N deﬁned by Y, (yn ) ∈ R ⇔ span{yn : n ∈ N} = Y is Borel. Now let A∗ = {Y ∈ SB : ∃X ∈ A with Y ∼ = X ∗ } be the dual class of A.

Of N such that one of the following (mutually exclusive) cases must occur: Case 1. The set {tn : n ∈ L} is an antichain. Our hypothesis in this case implies that for every n, m ∈ L with n = m the segments sn and sm are 0 incomparable. We deﬁne z = yl0 + . . + ylk0 . As the family (sli )ki=0 consists of pairwise incomparable segments of T , we get that k0 z ≥ Psli (z) i=0 2 1/2 (c) k0 = Psli (yli ) 2 1/2 (b) ≥ C k0 + 1. 15) i=0 Now we set w = z/ z ∈ Y . 14), for every segment s of T we have Ps (w) ≤ C 1 + k0 ε √ < .

P} with the property that there exists t ∈ Tjσ with t σ. For this particular jσ we have the trivial estimate Pσ (ujnσ ) ≤ 1 for every n ∈ N. Now ﬁx i ∈ {0, . . , p} with i = jσ . Then every node t in Ti is not an initial segment of σ. We deﬁne si = {σ|k : k ≥ 1} \ TGi = {σ|k : k ≥ 1} \ {τ |n : n ≥ 1 and τ ∈ Gi }. Notice, ﬁrst, that si is a ﬁnal segment of σ. Also notice that si is nonempty. Indeed, our assumption that every node t in Ti is not an initial segment of σ, simply reduces to the fact that si contains the ﬁnal segment {σ|k : k > l0 }.