A note about differentiability of maps defined on convex subsets of Banach spaces that may be nowhere dense (Q1378395)

The authors of this note consider differentiability of maps defined on convex subsets of Banach space that may be nowhere dense. In connection with continuum mechanics there are physically meaningful choices of infinite-dimensional Banach spaces such that the domain of constitutive maps is nowhere dense. The usual differential calculus is considered on open sets and not on nowhere dense sets. The authors give a differentiability for maps \(f\) defined on a convex subset of a Banach space which is not necessarily open. They define differentiability and prove theorems related to first and second derivatives of \(f\). A typical fact is the following: Let \(f: U\to F\) be differentiable on \(U\) and let \(a,b\in U\). Then \[ \| f(b)- f(a)\|\leq\| b- a\| \sup_{0\leq r\leq 1} \| f'((1- r)a+ rb)\|. \] Here \(U\) is a subset of function space and \(F\) be a Banach space.