Almost overcomplete and almost overtotal sequences in Banach spaces II

A sequence in a separable Banach space X < resp. in the dual space X*> is said to be overcomplete (OC in short) < resp. overtotal (OT in short) on X) whenever the linear span of each subsequence is dense in X (resp. each subsequence is total on X >. A sequence in a separable Banach space X < resp. in the dual space X*> is said to be almost overcomplete (AOC in short) < resp. almost overtotal (AOT in short) on X > whenever the closed linear span of each subsequence has finite codimension in X < resp. the annihilator (in X) of each subsequence has finite dimension). We provide information about the structure of such sequences. In particular it can happen that, an AOC < resp. AOT > given sequence admits countably many not nested subsequences such that the only subspace contained in the closed linear span of every of such subsequences is the trivial one < resp. the closure of the linear span of the union of the annihilators in X of such subsequences is the whole X >. Moreover, any AOC sequence {x(n)}(n is an element of N) contains some subsequence {x(nj)}(j is an element of N) that is OC in [{x(nj)}(j is an element of N)]; any AOT sequence {f(n)}(n is an element of N) contains some subsequence {n(j)}(j is an element of N) that is OT on any subspace of X complemented to {fnj}(j is an element of N)(T).