# Applications of Model Theory to Functional Analysis.pdf

Applications of Model Theoryto Functional AnalysisJos´e IovinoDepartment of MathematicsThe University of Texas at San AntonioSan Antonio, Texas USAXV Escuela Venezolana de Matem´aticasII Escuela Matem´aticas de Am´erica Latina y El Caribec©2002iiTo Martha and AbigailiiiFOREWORDThis text was commissioned by the Organizing Committee ofII Escuela de Matem´aticas de Am´erica Latina y el Caribe (IIEMALCA) and the XV Escuela Venezolana de Matem´aticas(XV EVM) to be used as textbook for the minicourse “Ul-traproductos en An´alisis”, one of four courses offered duringthe School, held September 8–14, 2002 at the Universidadde los Andes, M´erida, Venezuela.The author is most grateful to the Organizing Committee,and particularly to the Chair of the Committee, Dr. CarlosAugusto Di Prisco, for the kind invitation to conduct thecourse.The web site of the EMALCA/EVM School is:http://evm.ivic.ve.ContentsChapter 0. Introduction 1The Impact of Logic in Banach Space Theory 1The Case of Model Theory 2Model Theory for Structures of Functional Analysis 3Two Famous Applications 4On The Exposition 5Chapter 1. Preliminaries: Banach Space Models 71. Banach Space Structures and Banach Space Ultrapowers 72. Syntax: Positive Bounded Formulas 93. Semantics: Interpretations 114. Approximations of Formulas 135. Approximate Satisfaction 156. Beginning Model Theory 167. (1+epsilon1)-Isomorphism and (1+epsilon1)-Equivalence of Structures 198. Finite Representability 219. Types 2210. Quantifier-Free Types 2311. Saturated and Homogeneous Structures 2412. General Normed Space Structures 2713. The Monster Model 29Chapter 2. Semidefinability of Types 31Chapter 3. Maurey Strong Types and Convolutions 35Chapter 4. Fundamental Sequences 39Chapter 5. Quantifier-Free Types Over Banach Spaces 43Chapter 6. Digression: Ramsey’s Theorem for Analysis 47Chapter 7. Spreading Models 49vvi CONTENTSChapter 8. lscriptp- and c0-Types 51Chapter 9. Extensions of Operators by Ultrapowers 55Chapter 10. Where Does the Number p Come From? 57Chapter 11. Block Representability of lscriptp in Types 59Chapter 12. Krivine’s Theorem 63Chapter 13. Stable Banach Spaces 65Chapter 14. Block Representability of lscriptp in Types Over Stable Spaces 67Chapter 15. lscriptp-Subspaces of Stable Banach Spaces 69Chapter 16. Historical Remarks 73Bibliography 81Index of Notation 87Index 89CHAPTER 0IntroductionThe Impact of Logic in Banach Space TheoryIf one were to assemble a list of the most important results in of the lastthirty years in Banach space theory, the following would have to be included.(1) Tsirelson’s example of a Banach space not containing lscriptp or c0 [76];(2) Krivine’s Theorem [50];(3) The Krivine-Maurey theorem that every stable space contains somelscriptp almost isometrically [51];(4) The Bourgain-Rosenthal-Schechtman proof that there are uncount-ably many complemented subspaces of Lp [3];(5) Gowers’ dichotomy [22].Apart from their importance, these results have in common that theywere proved by borrowing concepts and techniques from logic. Tsirelson’sconstruction was inspired by the method of forcing; Krivine’s theorem wasproved using ultraproducts and compactness; the Krivine-Maurey theoremwas based on the notion of model theoretic stability; the main tool of theBourgain-Rosenthal-Schechtman paper is an ordinal rank of the type com-monly used in model theory; and Gowers dichotomy was proved using Gow-ers’ celebrated block Ramsey theorem [24], which is an elaboration of theGalvin-Prikry proof [17] of Silver’s theorem that every analytic set is Ram-sey [73].By all accounts, one of the most elegant theorems of modern Banachspace theory is Rosenthal’slscript1 theorem [65]. It was observed by Farahat [14]that Rosenthal’s proofcontainsanindependentproofofthe classical theoremthat every closed subset of NN is Ramsey. This observation unveiled infiniteRamsey theory as a chief tool in Banach space theory (for a rather oldbut excellent survey, see [55]) and triggered a host of applications thatculminated with Gower’s famous dichotomy [22].12 0. INTRODUCTIONThe Case of Model TheoryA close analysis of the concepts and techniques that have played animportant role in the development of modern Banach space theory will reveala striking number of them that are closely related to basic concepts frommodel theory. Examples are:(1) Indiscernible sequences (called 1-subsymmetric sequences in Ba-nach space theory);(2) Ordinal ranks (called ordinal indices in analysis);(3) Ehrenfeucht-Mostowski models (called spreading models in Banachspace theory);(4) Spaces of types;(5) Stability;(6) Ultrapowers.In some cases, these concepts have been introduced by adapting directlya construction from model theory to the context of Banach space theory(e.g., the case of Banach space ultrapowers, introduced by D. Dacunha-Castelle and J.-L. Krivine in [12]), in other cases, by analogy (e.g., thecase of Banach space stability, introduced by J.-L. Krivine and B. Maureyin [51]), and yet in other cases, concepts which are studied in model theory,as well as their connections with others, have been discovered independentlyby analysts (e.g., the case of indiscernible sequences — and their construc-tion using Ramsey’s Theorem — which were introduced by A. Brunel andL. Sucheston in the study of ergodic properties of Banach spaces; see [5]).In addition, some concepts that play a central role in Banach spacetheory (e.g., that of finite representability) can be seen naturally as modeltheoretical phenomena (a Banach spaceX is finitely represented in a BanachspaceY if and only ifY is a model of the existential theory ofX). There areeven similarities between classification programs in both fields. For example,the dichotomy reflexive/unreflexive in Banach space theory is equivalent,in a categorical sense, to the dichotomy stable/unstable in model theory.(See [39].) Also, in both fields, the role played by partition theorems isregarded as fundamental.These phenomena suggest that the relation between the two fields israther deep. Given the remarkable technical complexity that both fieldshave attained in the last thirty years, we suggest that it would be desirableto have clearly understood channels of communication between them sothat techniques from one field might become useful in the other. Someconsiderations are in order, however.MODEL THEORY FOR STRUCTURES OF FUNCTIONAL ANALYSIS 3(1) First-order logic is not the natural logic to analyze Banach spaces asmodels. Banach space theory is carried out in higher order logics,as is functional analysis in general. Furthermore, the first-ordertheory of Banach spaces is known to be equivalent to a secondorder logic. (See [72].)(2) The concepts from Banach space theory listed above are not theliteral translations of their first-order analogs. For instance, a Ba-nach space ultrapower of a Banach space X is not an ultrapowerof X in the sense customarily considered in model theory, and isnot an elementary extension of X in the sense of first-order logic.However, there is a strong analogy between the role played by Ba-nach space ultrapowers in Banach space theory and that played byalgebraic ultrapowers in model theory.Let us illustrate this point with a second example. What isregarded in Banach space theory as the “space of types” is notwhat is understood as the space of types in the first-order sense.Let us recall definition given in [51]:Let X be a fixed separable Banach space. A typeon X is a function τ: X →R such that there existsa sequence (xn) in X satisfyingτ(x) = limn→∞bardblx+xnbardbl.The space of types of X, as defined in [51], is the set of typeson X with the topology of pointwise convergence. This notion ofspace of types is motivated by the corresponding notion from first-order logic. The analogy is not entirely clear a priori. However,as we shall see, both notions of type are connected by a naturalinterpretation.Model Theory for Structures of Functional AnalysisA formal framework for a model theoretical analysis of Banach spaceswas first introduced by C. W. Henson in [35]; the scope expanded by Hensonand the author in [36]. Although this framework was originally introducedfor Banach spaces, it can be generalized naturally to include rich classes ofstructures from functional analysis. The unique feature of this logical ap-proach is that, although it is appropriate for structures from functional anal-ysis, it preserves many of the desirable characteristics of first-order modeltheory, e.g., the compactness theorem, L¨owenheim-Skolem theorems, and4 0. INTRODUCTIONomitting types theorem. (In fact, it provides a natural setting for the clas-sification theory, in the sense of [71], of structures from infinite dimensionalanalysis.) Furthermore, it provides a uniform foundation for the contribu-tions mentioned above. For example, the role played by analytic ultrapowersin this framework mirrors that played by algebraic ultrapowers in first-ordermodel theory; also, types in the sense of [51] described above correspondexactly to quantifier-free types in this context, indiscernibles in the senseof [5] are quantifier-free indiscernibles, and the kind of Banach space stabil-ity introduced in [51] corresponds exactly to quantifier-free stability of thestructure.Two Famous ApplicationsThe problem of how the classical sequence spaces lscriptp (1 ≤p 0, monadic predicates for the sets{x∈X | bardblxbardbl ≤M} and {x∈X | bardblxbardbl ≥M};· A monadic function symbol (an operator symbol) for each operatorTj;· A syntactic symbol (a constant symbol) for each element ck;· Upper norm bounds for each element ck and each operator Tj.We say that X is a Banach space L-structure, or simply, an L-structure.1.1. Remark. For every language L, the class of L-structures is closedunder ultraproducts. (Notice that the requirement that the language includebounds for each constant and operator symbols is needed for this.)10 1. PRELIMINARIES: BANACH SPACE MODELSFix a language L for a Banach space structure. We now define sets ofstrings of symbols called the terms and positive bounded formulas ofL. Bothdefinitions are recursive.A term of L (or an L-term) is any string of symbols which can beobtained by finitely many applications of the following rules of formation:· If x is a syntactic variable, then x is an L-term.· 0 (the syntactic symbol for the additive identity ofX) is anL-term.· If c is a constant symbol of L, then c is an L-term.· If t1 and t2 are L-terms, then t1 +t2 is an L-term.· If r is a symbol for scalar multiplication, and t is an L-term, thenr·(t) is an L-term.· If t is an L-term and T is an operator symbol of L, then T(t) is anL-term.A positive bounded formula of L (or a positive bounded L-formula) is astring of symbols which can be obtained by finitely many applications of thefollowing rules of formation:· If t is an L-term and M is a positive rational number, then theexpressionsbardbltbardbl ≤M, bardbltbardbl ≥Mare positive bounded L-formulas.· If ϕ1 and ϕ2 are positive bounded L-formulas, then the expressions(ϕ1 ∧ϕ2), (ϕ1 ∨ϕ2)are positive bounded L-formulas.· If ϕ is a positive bounded L-formula, x is a variable, and M is apositive rational number, then the expressions∃x(bardblxbardbl ≤M ∧ϕ),∀x(bardblxbardbl ≤M →ϕ)are positive bounded L-formulas.Thus, a positive bounded formula is an expression built up from theatomic formulasbardbltbardbl ≤M, bardbltbardbl ≥M(where t is a term of L and M 0) by using the positive connectives ∧,∨and the bounded quantifiers∃x(bardblxbardbl ≤M ∧ . ) and ∀x(bardblxbardbl ≤M → . )(where M 0).3. SEMANTICS: INTERPRETATIONS 11If t is a term and M1,M2 are real numbers, we write M1 ≤ bardbltbardbl ≤ M2as an abbreviation of the positive bounded formula (M1 ≤ bardbltbardbl∧bardbltbardbl ≤M2).Similarly, we write bardbltbardbl = M as an abbreviation of (M ≤ bardbltbardbl∧bardbltbardbl ≤ M).Often, when the context allows it, we omit the outer parentheses in formulasof the form (ϕ1 ∧ϕ2) or (ϕ1 ∨ϕ2). Sometimes we also write logicalandtextni=1ϕi andlogicalortextni=1ϕi as abbreviations of ϕ1 ∧···∧ϕn and ϕ1 ∨···∨ϕn, respectively.If ϕ is a positive bounded formula, a subformula of ϕ is any string ofconsecutive symbols of ϕ which is a positive bounded formula in its ownright.A variable x is said to occur free in a positive bounded formula ϕ ifx occurs in ϕ and is not under the scope of any of the quantifiers thatoccur in ϕ (i.e., there is at least one occurrence of x in ϕ which is notwithin any subformula of ϕ of the form ∃xψ or ∀xψ.) A positive boundedsentence is a positive bounded formula without free variables. If t is a termand x1,.,xn are variables, we write t(x1,.,xn) to indicate that all thevariables occurring in t are among x1,.,xn. Similarly, if ϕ is a positivebounded formula, we write ϕ(x1,.,xn) to indicate that all the variablesthat occur free in ϕ are among x1,.,xn.3. Semantics: InterpretationsSuppose thatX = ( X, Tj, ck | j ∈J, k ∈K )is a Banach space L-structure and t(x1,.,xn) is an L-term. If a1,.,anare elements of X, we denote bytX[a1,.,an],the element ofX that results from interpreting the variablexi as the elementai, for i = 1,.,n. The formal definition is by induction on the complexityof t, as follows:· If t(x1,.,xn) is the variable xi (i = 1,.,n), then tX[a1,.,an]is the element ai.· If t is 0, then tX[a1,.,an] is the additive identity of X.· If t(x1,.,xn) is the constant symbol c, then tX[a1,.,an] is theinterpretation of c in X.· If t(x1,.,xn) is t1(x1,.,xn) + t1(x1,.,xn), where t1 and t2are L-terms, thentX[a1,.,an] = tX1 [a1,.,an]+tX1 [a1,.,an].12 1. PRELIMINARIES: BANACH SPACE MODELS· If t(x1,.,xn) is r· (t1(x1,.,xn)), where t1 is an L-term and ris a syntactic symbol for scalar multiplication, thentX[a1,.,an] = r(tX[a1,.,an]).· If t(x1,.,xn) is T(t1(x1,.,xn)), where where T is a syntacticsymbol for the operator Tj and t1 is an L-term, thentX[a1,.,an] = Tj(tX1 [a1,.,an]).Note that tX[a1,.,an] depends not only on t, X, and a1,.,an, butalso on a given list of variables which is not given explicitly by the notation.The context will normally make it it clear which list of variables is beingconsidered.If ϕ(x1,.,xn) is an L-formula and a1,.,an are elements of X, wewriteX |= ϕ[a1,.,an]if the formula ϕ is true in X when the variable xi is interpreted as theelement ai, for i = 1,.,n. This concept should be intuitively clear. Theformal definition is by induction on the complexity of ϕ, as follows:· If ϕ is t(x1,.,xn) ≤M, X |= ϕ[a1,.,an] if and only iftX[a1,.,an] ≤M.· If ϕ is t(x1,.,xn) ≥M, X |= ϕ[a1,.,an] if and only iftX[a1,.,an] ≥M.· If ϕ is (ψ1 ∧ψ2), then X |= ϕ[a1,.,an] if and only ifX |= ψ1[a1,.,an] and X |= ψ2[a1,.,an].· If ϕ is (ψ1 ∨ψ2), then X |= ϕ[a1,.,an] if and only ifX |= ψ1[a1,.,an] or X |= ψ2[a1,.,an].· If ϕ is ∃x(bardblxbardbl ≤ M ∧ ψ(x,x1,.,xn)), where M is a positiverational number and x is a variable, then X |= ϕ[a1,.,an] if andonly ifX |= ψ[a,a1,.,an], for some a∈X with bardblabardbl ≤M.· If ϕ is ∀x(bardblxbardbl ≤ M → ψ(x,x1,.,xn)), where M is a positiveratio