Unconditional basic sequence

An unconditional basic sequence is a Schauder basis for its closed span whose expansions converge after arbitrary permutations or coordinate suppressions. Equivalently, the finite coordinate projections are uniformly bounded. Failure of this property gives pairs of disjointly supported unit vectors at arbitrarily small distance, obtained by separating a finite combination with a large coordinate projection.

Let $T_C(X)$ consist of finite normalized sequences for which every scalar combination and sign change satisfy $\Vert {\sum }_i{\epsilon }_i{a}_i{x}_i\Vert \le C\Vert {\sum }_i{a}_i{x}_i\Vert$. These inequalities give a closed tree on a Polish space when $X$ is separable. An infinite branch is an unconditional basic sequence. If there is no such sequence, each fixed-constant tree is well-founded and has countable rank of a well-founded tree. Consequently, finite unconditional configurations with one fixed constant and arbitrarily high countable ranks force an infinite unconditional sequence.

A basic sequence is suppression-unconditional with constant $C$ if every coordinate deletion satisfies the displayed inequality. When $C=1$, deletion never increases the norm. Uniformly bounded finite-coordinate projections imply unconditional convergence of the basis expansions; for real scalars, the sign-change constant is at most $2C$.