Generalized projections, decompositions, and the Pythagorean-type theorem in Banach spaces
Let \(B\) be a real uniformly convex and uniformly smooth (reflexive) Banach space, \(B^*\) its conjugate space, \(J:B\to B^*\) the duality mapping determined by the equalities \(\langle Jx,x\rangle = \|Jx\|\|x\|=\|x\|^2,\) and let \(W(x,\xi) = \|x\|^2 - 2\langle Jx,x \rangle + \|\xi\|^2 \). The functional \(W\), called the Lyapunov functional, is positive and it is closely related to the Young-Fenchel transform. In [Lect. Notes Pure Appl. Math. 178, 15-50 (1996; Zbl 0883.47083)], the author defined and studied the properties of generalized projection operators \(\Pi_\Omega\) and \(\pi _\Omega=J^*\Pi_\Omega\) where, for a closed convex subset \(\Omega\) of \(B\), \(\Pi_\Omega\) is defined by \(\Pi_\Omega = \widehat x,\quad \widehat x\) being the solution of the minimization problem \(W(x,\widehat x) =\inf _{\xi \in \Omega} W(x,\xi).\) If \(B\) is a Hilbert space then \(Jx = x\), \(W(x,\xi) = \|x-\xi\|^2\) and \(\Pi_\Omega\) becomes the usual metric projection operator on \(\Omega.\) As it was shown by the author, \textit{A. N. Iusem} and \textit{M. V. Solodov} [J. Convex Anal. 4, No. 2, 235-255 (1997; Zbl 0895.90150)], the generalized projection operators have nice properties which make them very effective, in contrast to the metric projection operators, in the study of approximation and optimization problems in Banach spaces. The aim of the present paper is to study the properties of the generalized projection operators in the particular case when \(\Omega\) is a closed convex cone \(K\) of the Banach space \(B\). One proves decomposition theorems for arbitrary elements, both in primary and dual spaces, and one establishes Pythagorean-type rules, \(\|x\|^2 = \|\widehat x\|^2 + W(x,\widehat x),\) results which were known previously only in Hilbert spaces.
scientific article; zbMATH DE number 1369487