Approximation scheme in a Banach space with decomposition (Q2566855)

Let \(E\) be a Banach space over the field \(\mathbb{K}\) of real or complex numbers. For each \(n\in\mathbb{N}^*= \mathbb{N}\cup\{0\}\), let \(Q= \{Q_n(E): n\in\mathbb{N}^*\}\) be a sequence of families of subsets of \(E\) satisfying (i) \(\{0\}= Q_0(E)\subset Q_1(E)\subset\cdots\subset Q_h(E)\subset\cdots\), (ii) If \(A\in Q_n(E)\), \(\lambda\in\mathbb{K}\) then \(\lambda A\in Q_n(E)\forall n\in\mathbb{N}^*\), (iii) If \(A\in Q_n(E)\), \(B\in Q_m(E)\) then \(A+ B\in Q_{n+m}(E)\) for every \(n,m\in\mathbb{N}^*\). Then \(Q= \{Q_n(E): n\in\mathbb{N}^*\}\) is called an approximation scheme on \(E\). An infinite sequence \(\{M_n\}\) of nonzero line are subspaces of \(E\) is called a decomposition of \(E\) if for every \(x\in E\), there exists a unique sequence \(\{y_n\}\subset E\) with \(y_n\in M_n\) \((n=1,2,\dots)\) such that \(x= \sum^\infty_{i=1} y_i\), the convergence being in the norm topology of \(E\). In this paper, the authors construct an approximation scheme in \(E\) with a Schauder decomposition \(\{M_n\}\) by the family \({\mathcal F}(E)\) of subsets \({\mathcal F}_n(E)= \bigcup_\xi \{\sum_{i\in\xi}\oplus M_i:\) and \(\xi\leq n\}\), of all at most \(n\)-superstable elements, i.e., \(\sum_{i\in\xi}\oplus M_i= \{x: x= \sum_{i\in\xi} x_i,\, x_i\in M_i\}\), with \({\mathcal F}_0(E)= \{0\}\). By means of this approximation scheme, they define for every element \(x\in E\) a sequence \(\{\alpha_n(x): n\in\mathbb{N}^*\}\) of approximation numbers as \(\alpha_n(x)= \inf\{\| x- a\|: a\in\sum_{i\in\xi}\oplus M_i\in{\mathcal F}_n(E)\}\) and study their properties (Propositions 1 and 2). They also study some properties of a subclass of compact sets of type \(\ell^p\) whose sequence of diameters of their projections on the decomposing subspaces is \(p\)-summable. Certain characterizations of the spaces \(S^p\) of elements of type \(\ell^p\) whose sequence of approximation numbers is \(p\)-summable have also been obtained in this paper.