Block sequence

Relative to a Schauder basis, a block sequence consists of nonzero finitely supported vectors with successive disjoint supports: $\max \mathrm{supp}{x}_i<\min \mathrm{supp}{x}_{i+1}$. It is a basic sequence, with basis constant at most that of the original basis.

A block subspace is the closed linear span of an infinite block sequence. Passing to a block subspace preserves the order of supports and permits successive-vector constructions.

If an analytic family of finite normalized block sequences meets every infinite-dimensional block subspace, then after an arbitrarily prescribed coordinatewise perturbation it is strategically large in some block subspace. In the block game, one player chooses an infinite-dimensional block subspace at each turn and the other chooses a successive unit vector from it; strategically large means the vector player can force a finite initial segment into the family.