We construct a hereditarily indecomposable Banach space with dual space isomorphic to ℓ1. Every bounded linear operator on this space is expressible as λI + K with λ a scalar and K compact.

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Certain subclasses of B 1 (K), the Baire-1 functions on a compact metric space K, are defined and characterized. Some applications to Banach spaces are given. Introduction.Let X be a separable infinite dimensional Banach space and let K denote its dual ball, Ba(X * ), with the weak* topology. K is compact metric and X may be naturally identified with a closed subspace of C(K). X * * may also be identified with a closed subspace of

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