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∈N contains some subsequence {x_nj}j∈N that is OC in [{x_nj}j∈N]; any AOT sequence {f_n}n∈N contains some subsequence {f_nj}j∈N that is OT on any subspace of X complemented to {f_nj}j∈N⊤.