The linear generalized gradient in infinite dimensions (Q5956093)

The author extends the linear generalized gradient of lower semicontinuous functions (defined former for finite-dimensional spaces) to general Banach spaces \(X\). The definition bases on a family \({\mathfrak Y}\) of (e.g., finite-dimensional) linear subspaces \(Y\subset X\) withNEWLINENEWLINE\hskip 17mm (1) \(X= \bigcup_{Y\in{\mathfrak Y}} Y\),NEWLINENEWLINE\hskip 17mm (2) \(\forall Y_1,Y_2\in{\mathfrak Y} \exists Y\in{\mathfrak Y}: Y_1+ Y_2\subset Y\),NEWLINENEWLINE\hskip 17mm (3) \(\forall Y\in{\mathfrak Y} \exists\) renorm \(\|\cdot\|\) such that \(\|\cdot\|^2\) is differentiable on \(Y\)NEWLINENEWLINE\hskip 17mm\qquad \qquad with locally Lipschitz derivative.NEWLINENEWLINENEWLINENow a linear functional \(v^*\in X^*\) is called to be a \(Y\)-partial viscosity subderivative (pvsd) of \(f: X\to \mathbb{R}\) at the point \(x\in X\) if there exists a Lipschitz convex function \(g: X\to\mathbb{R}\) such thatNEWLINENEWLINE\hskip 17mm (1) \(-v^*\in\partial g(x)\),NEWLINENEWLINE\hskip 17mm (2) \(g\) is \(Y\)-smooth,NEWLINENEWLINE\hskip 17mm (3) \(g+f\) has a local minimum at \(x\).NEWLINENEWLINENEWLINEFinally, the linear generalized gradient of \(f\) at \(\overline x\) is introduced through a limiting process using \(Y\)-linear nets (sequences) of \(Y\)-partial viscosity subderivatives according to NEWLINE\[NEWLINE\partial_\ell f(x)= \bigcap_{Y\in{\mathfrak Y}} \text{cl}^*\{v^*\mid v^*\text{ is the weak\(^*\) limit of a \(Y\)-linear net of pvsd}\}.NEWLINE\]NEWLINE It is shown -- as in finite dimensions -- that the linear generalized gradient is contained in Ioffe's \(G\)-derivative, Clarke's generalized gradient and Mordukhovich's subderivative. If \(f\) is Lipschitz, then \(\partial_\ell f(x)\neq \emptyset\); if \(f\) is Fréchet differentiable, then \(\partial_\ell f(x)= \{\nabla f(x)\}\).NEWLINENEWLINENEWLINEThe second part of the paper is devoted to basic calculus rules and a simple Lagrange multiplier rule.