Using a geometric method, we characterize all entire functions that transform the Bloch space into a Bergman space by superposition in terms of their order and type. We also prove that all superposition operators induced by such entire functions act boundedly. Similar results hold for superpositions from BMOA into Bergman spaces and from the Bloch space into certain weighted Hardy spaces.

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We show that if$E$is a complex Banach space which contains no subspace isomorphic to$l$_{1}, then each infinite dimensional subspace of$E′$contains a normalized sequence which converges to zero for the weak star topology.

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