Neocompact quantifier elimination in structures based on Banach spaces

Consequences of neocompact quantifier elimination

“…In our previous paper [4] we have introduced a property of Quantifier Elimination (for short: QE) in the setting of Banach space structures (certain first order structures based on Banach spaces that will be described in Section 2). Property QE is a reformulation of a notion introduced in [20].…”

“…In [4] we studied the law structure associated to a Banach space structure. We observed that atomic neocompact formulas become countable conjunctions of quantifier-free positive bounded formulas (a class of formulas introduced in [10] by Henson in order to carry out a model-theoretic investigation of Banach spaces).…”

“…We observed that atomic neocompact formulas become countable conjunctions of quantifier-free positive bounded formulas (a class of formulas introduced in [10] by Henson in order to carry out a model-theoretic investigation of Banach spaces). More in detail, in [4] we specialized properties of the law structures to the class of law structures that are associated to Banach space structures and we formulated our QE property. We also proved that if the nonstandard hull of a Banach space structure has QE, such a property is independent of the nonstandard universe where the nonstandard hull has been constructed.…”

“…In this follow up of [4], we investigate whether some well-known classes of Banach spaces have QE. We also investigate whether QE or weaker properties have consequences that are of interest to functional analysis.…”

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Key words Banach space structures, quantifier elimination, neocompact, positive bounded formula, nonstandard hull.

MSC (2000)03C10, 03H05, 03C65, 46B08, 46B04We provide some consequences of a Quantifier Elimination Property and related properties previously introduced (see [4]) in the setting of Banach space structures. We further consider some applications of quantifierfree definability, such as strict convexity via the definability of certain mapping and the continuity of definable functions.

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