Basis Properties of Some Systems In Banach Spaces

Abstract

Let uˆn = (un, an), n = 1, 2, ... be some complete and minimal system of vectors in X = X 0 ⊕ C m and let ϑˆ n = (ϑn, bn), n = 1, 2, ... be corresponding biorthogonal system. N is a set of natural numbers, J = {n1, ..., nm} ⊂ N is some set of different and natural numbers, n0 = N \J, bn = (βn1 , ..., βnm), δ = det

βnkj

m k,j=1. In the present paper it is shown that in case of δ = 0 statement on non-minimality of the system {un}n∈N0 in the space X0, in generally, is not true, and sufficient conditions are cited when this statement becomes true.