John’s Decompositions: Selecting a Large Part

We extend the invertibility principle of J. Bourgain and L. Tzafriri to operators acting on arbitrary decompositions id = P xj ⊗xj , rather than on the coordinate one. The John’s decomposition brings this result to the local theory of Banach spaces. As a consequence, we get a new lemma of Dvoretzky-Rogers type, where the contact points of the unit ball with its maximal volume ellipsoid play a crucial role. We then apply these results to embeddings of l ∞ into finite dimensional spaces.