Essential supremum norm differentiability

Abstract

The points of Gateaux and Fréchet differentiability in L∞(μ,X) are obtained, where (Ω,∑,μ) is a finite measure space and X is a real Banach space. An application of these results is given to the space B(L1(μ,ℝ),X) of all bounded linear operators from L1(μ,ℝ) into X.

Keywords

• Banach spaces
• measurable functions
• essentially bounded functions
• vector-valued functions
• essential supremum norm
• smooth points.