Characterizations of the metric and generalized metric projections on subspaces of Banach spaces (Q6542916)

The authors investigate interesting types of projection onto the subspaces of general Banach spaces. To be precisely, let \(B:=(B,\|\cdot\|)\) be a Banach space with the dual space \(B^*:=(B^*,\|\cdot\|_*)\) and let \(\varnothing\neq K\subset B\). The metric projection \(P_K\) is given by \(P_K(x):=\{y\in K:\|x-y\|=\min_{z\in K}\|x-z\|\}\). The generalized projection from \(B^*\) to \(2^K\) is given by \(\pi_K(\varphi):=\{y\in K:V(\varphi,y)=\min_{y\in K}V(\varphi,y)\}\) where \(V(\varphi,x):=\|\varphi\|_*^2-2\langle\varphi,x\rangle+\|x\|^2\) and \(\langle\varphi,x\rangle\) is the pairing between \(B^*\) and \(B\). The generalized metric projection from \(B\) to \(2^K\) is given by \(\Pi_K(x):=\bigcup_{j\in Jx}\pi_K(j)\) where \(Jx:=\{j\in B^*:\langle j,x\rangle=\|j\|_*\|x||=\|x\|^2\}\). The variational principles for the metric projection and the generalized metric projection on subspaces of Banach spaces are given. They also present a concrete discussion on certain subspaces of the sequence spaces \(c\) and \(\ell_1\); and the function space \(C[0,1]\).