Abstract
We extend the invertibility principle of J. Bourgain and L. Tzafriri to operators acting on arbitrary decompositionsid = ∑x j ⊕x j , 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 ofl k∞ into finite dimensional spaces.
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Vershynin, R. John's decompositions: Selecting a large part. Isr. J. Math. 122, 253–277 (2001). https://doi.org/10.1007/BF02809903
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DOI: https://doi.org/10.1007/BF02809903