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In this paper we settle in the negative the problem of the superreflexivityof Garling sequence spaces by showing that they contain a complemented subspaceisomorphic to a non superreflexive mixed-norm sequence space. As a by-productof our work, we give applications of this result to the study of conditionalSchauder bases and conditional almost greedy bases in this new class of Banachspaces.

Non-superreflexivity of Garling sequence spaces and applications to the existence of special types of conditional bases