Basic Linear Structure

A sequence {e i }i = 1 ∞ in a Banach space X is called a Schauder basis for X if for each x ∈ X there is a unique sequence of scalars {α i }i = 1 ∞ such that \(x =\sum _{ i=1}^{\infty }\alpha _{i}e_{i}\). If the convergence of this series is unconditional for all x ∈ X (i.e., any rearrangement of it converges), we say that the Schauder basis is unconditional . This is equivalent to say that under any permutation \(\pi: \mathbb{N} \rightarrow \mathbb{N}\), the sequence {eπ(i)}i = 1 ∞ is again a basis of X.