Let X and Y be quasi complementary subspaces of a separable Banach space B and let (zn) be a sequence complete in X. Then (a) there exists a uniformly minimal norming M-basis (xn ) of X with xm C span(zn)n>q, for every m, qm -4 oc; (b) if (xn) is a uniformly minimal norming M-basis of X, there exists a uniformly minimal norming M-basis of B which is an extension of (xn); (c) there exists a uniformly minimal norming M-basis (xn) U (Yn ) of B with (xn) C X and (Yn) C Y. Introduction. The problem of the best complete sequence in a separable Banach space appears already in Banach's book [1]. The story of this problem goes through the existence of the M-basis (Markushevich [7], 1943), the existence of the norming M-basis (Mackey [63, 1946) and other intermediate results (DavisJohnson [2], 1973); it culminates with a negative answer to the basis problem (Enflo [3], 1973) and with the existence of the uniformly minimal norming M-basis (Ovsepian and Pelczynski [9], 1975; refined by Pelczynski [10], 1976). The aim of this note is to study particular constructions of uniformly minimal norming M-bases. In this direction three types of questions appear: (i) The regularization of a sequence (xn) of B with [Xn] of infinite dimension; indeed, by [9] there exists a uniformly minimal norming M-basis (Zn) of [Xn], but without any connection with (xn) more than the spanned space is the same. If (Pn) is a sequence of positive integers let us say that (yn) is (pm)-ordered in (xn) if Ym E span(xn)n>pm with Pm+1 > Pm for every m, moreover limm+o Pm = 00. Then the following connection is possible. THEOREM I. For every (Xn) in a Banach space, with [Xn] of infinite dimension, there exists a 1-norming uniformly minimal M-basis (Zn) of [Xn], which is (Pm)> ordered in (Xn). This construction is considered in ?1. (ii) The extension in a given direction: if (xn) is a uniformly norming M-basic sequence of a separable Banach space B, is it possible to extend (xn) to a uniformly minimal norming M-basis of B? In particular is it possible to effect the extension by adding elements of a subspace Y of B which is quasi complementary with [xn]? Received by the editors February 21,1984 and, in revised form, May 15, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 46B30. (0)1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page

https://www.ams.org/tran/1987-299-02/S0002-9947-1987-0869217-4/S0002-9947-1987-0869217-4.pdf

On the theory of fundamental norming bounded biorthogonal systems in Banach spaces

Viene presentato un panorama della ricerca della successione con le migliori proprieta che possa rappresentare uno spazio di Banach separabile, considerando in particolare l’estensione di una successione uniformemente minimale e normante e la costruzione in date direzioni di unaM-base uniformemente minimale e normante. Inoltre viene presentato un panorama della ricerca della successione infinita con le migliori proprieta presente in ogni spazio di Banach separabile, considerando in particolare il problema della successione basica incondizionale.

Sistemi biortogonali negli spazi di Banach

We prove Milman's theorem on the extension, in a given direction, of M-basic sequence to M-basis in a separable Banach space.

http://yadda.icm.edu.pl/yadda/element/bwmeta1.element.baztech-article-AGH4-0006-0008/c/Plichko.pdf

Paolo Terenzi

Papers

On bounded and total biorthogonal systems spanning given subspaces