All separable Banach space admit for every ε>0 fundamental total and bounded by 1 + ε biorthogonal sequences

Disjoint mixing operators

“…According to Pelczynski [23], we can pick sequences {x n } n∈Z and {f n } n∈Z in X and X * respectively such that span{x n : n ∈ Z} is dense in X, span{f n } n∈Z is dense in X * , f n (x m ) = δ n,m for each m, n ∈ Z and x n 2, f n 2 for each n ∈ Z. By Corollary 4.2, there is a…”

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

“…The story of this problem goes through the existence of the M-basis (Markushevich [7], 1943), the existence of the norming M-basis (Mackey [6], 1946) and other intermediate results (Davis-Johnson [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).…”

Minimal ball-coverings in Banach spaces and their application

“…Without loss of generality we assume that X is nonseparable; otherwise, X itself has the ball-covering property. Fix any separable infinitedimensional closed subspace X 0 ⊂ X, and for every 0 < ε < 1, applying a theorem of Pełczyński [14] to X 0 , we find that there exists a normalized biorthogonal system {(…”

“…Almost all properties of Banach spaces, such as convexity, smoothness, reflexivity, the Radon-Nikodým property, etc., can be viewed as properties of the unit ball. We should also mention here several topics concerning the behavior of families of balls, for example, the Mazur intersection property (see, for instance, [14] Starting from a different viewpoint, this article is devoted to studying the behavior of families B of open balls off the origin in a Banach space X whose union contains the unit sphere of X. We call such a family B a ballcovering of X.…”

“…Almost all properties of Banach spaces, such as convexity, smoothness, reflexivity, the Radon-Nikodým property, etc., can be viewed as properties of the unit ball. We should also mention here several topics concerning the behavior of families of balls, for example, the Mazur intersection property (see, for instance, [14], [16], [17]), the sphere packing problem (see, for instance, [7] and [13]), and the measure of non-compactness with respect to topological degree (see, for instance, [1], [2], [10]), which have also attracted attention of many mathematicians.…”