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.

We study the relation between different spaces of vector-valued polynomials and analytic functions over dual-isomorphic Banach spaces. Under conditions of regularity on$E$and$F$, we show that the spaces of$X$-valued$n$-homogeneous polynomials and analytic functions of bounded type on$E$and$F$are isomorphic whenever$X$is a dual space. Also, we prove that many of the usual subspaces of polynomials and analytic functions on$E$and$F$are isomorphic without conditions on the involved spaces.

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