Γ-Null and ΓN-Null Sets
Γ-Null and ΓN-Null Sets
This chapter introduces the notions of Γ-null and Γₙ-null sets, which are σ-ideals of subsets of a Banach space X. Γ-null set is key for the strongest known general Fréchet differentiability results in Banach spaces, whereas Γₙ-null set presents a new, more refined concept. The reason for these notions comes from an (imprecise) observation that differentiability problems are governed by measure in finite dimension, but by Baire category when it comes to behavior at infinity. The chapter first relates Γ-null and Γₙ-null sets to Gâteaux differentiability before discussing their basic properties. It then describes Γ-null and Γₙ-null sets of low Borel classes and presents equivalent definitions of Γₙ-null sets. Finally, it considers the separable determination of Γ-nullness for Borel sets.
Keywords: null sets, Banach space, Fréchet differentiability, Gâteaux differentiability, low Borel classes, separable determination, Borel sets
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