Gowers Ramsey theorem for Banach spaces

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.