Separable reduction theorems by the method of elementary submodels

Marek Cúth¹

¹ Faculty of Mathematics and Physics Charles University Sokolovská 83 186 75 Praha 8 Karlín, Czech Republic

Fundamenta Mathematicae, Tome 219 (2012) no. 3, pp. 191-222.

Résumé

We simplify the presentation of the method of elementary submodels and we show that it can be used to simplify proofs of existing separable reduction theorems and to obtain new ones. Given a nonseparable Banach space $X$ and either a subset $A\subset X$ or a function $f$ defined on $X$, we are able for certain properties to produce a separable subspace of $X$ which determines whether $A$ or $f$ has the property in question. Such results are proved for properties of sets: of being dense, nowhere dense, meager, residual or porous, and for properties of functions: of being continuous, semicontinuous or Fréchet differentiable. Our method of creating separable subspaces enables us to combine results, so we easily get separable reductions of properties such as being continuous on a dense subset, Fréchet differentiable on a residual subset, etc. Finally, we show some applications of separable reduction theorems and demonstrate that some results of Zajíček, Lindenstrauss and Preiss hold in the nonseparable setting as well.

Détail

DOI : 10.4064/fm219-3-1

Keywords: simplify presentation method elementary submodels simplify proofs existing separable reduction theorems obtain given nonseparable banach space either subset subset function defined able certain properties produce separable subspace which determines whether has property question results proved properties sets being dense nowhere dense meager residual porous properties functions being continuous semicontinuous chet differentiable method creating separable subspaces enables combine results easily get separable reductions properties being continuous dense subset chet differentiable residual subset etc finally applications separable reduction theorems demonstrate results zaj lindenstrauss preiss nonseparable setting