Definable norms and definable types over Banach spaces
نویسندگان
چکیده
A central question in Banach space theory has been to identify the class of Banach spaces that contain almost isometric copies of the classical sequence spaces `p and c0. Banach space theory entered a new era in the mid 1970’s, when B. Tsirelson [34] constructed the first space not containing isomorphic copies any of the classical sequence spaces. Tsirelson’s space has been called “the first truly nonclassical Banach space” [8]. The greatest innovation in Tsirelson’s two-page paper is that the norm of the space is not defined explicitly, by a formula, but implicitly, by an equation; the norm appears on both sides of the equation, so at first sight the definition appears to be circular; a fixed point argument shows that such a norm exists. After Tsirelson’s construction, the following question arose in the folklore of Banach space theory.
منابع مشابه
Stable Banach Spaces and Banach Space Structures, I: Fundamentals
We study model theoretical stability for Banach spaces and structures based on Banach spaces, e.g., Banach lattices or C∗-algebras. We prove that a theory is stable if and only if the following condition is true in every model E of the theory: If (ām ) and (b̄n) are bounded sequences in Ek and El (respectively) and R : Ek × El → R is definable, then there exist subsequences (āmi ) and (b̄n j ) su...
متن کاملOn the Topology of Metric Spaces Definable in o-minimal expansions of fields
We study the topology of metric spaces which are definable in o-minimal expansions of ordered fields. We show that a definable metric space either contains an infinite definable discrete set or is definably homeomorphic to a definable set equipped with its euclidean topology. This implies that a separable metric space which is definable in an o-minimal expansion of the real field is definably h...
متن کاملDefinable Subsets in Covering Approximation Spaces
Covering approximation spaces is a class of important generalization of approximation spaces. For a subset X of a covering approximation space (U, C), is X definable or rough? The answer of this question is uncertain, which depends on covering approximation operators endowed on (U, C). Note that there are many various covering approximation operators, which can be endowed on covering approximat...
متن کاملOn metric types that are definable in an o-minimal structure
In this paper we study the metric spaces that are definable in a polynomially bounded ominimal structure. We prove that the family of metric spaces definable in a given polynomially bounded o-minimal structure is characterized by the valuation field Λ of the structure. In the last section we prove that the cardinality of this family is that of Λ. In particular these two results answer a conject...
متن کاملOn Decidability of T-norm-Based Equational Theories
The aim of this work is to show that the universal theory R of real closed fields [1] is interpretable in the equational theory of LPi1/2-algebras [2,3,6,7,8], and viceversa. Since R enjoys quantifier elimination, we will obtain that the full theory of R is interpretable in LPi1/2. This will also yield that any function definable in R is definable in LPi1/2. As a consequence of this constructio...
متن کامل