Memoirs of the American Mathematical Society 2004; 114 pp; softcover Volume: 170 ISBN10: 0821835211 ISBN13: 9780821835210 List Price: US$65 Individual Members: US$39 Institutional Members: US$52 Order Code: MEMO/170/806
 A general method producing Hereditarily Indecomposable (H.I.) Banach spaces is provided. We apply this method to construct a nonseparable H.I. Banach space \(Y\). This space is the dual, as well as the second dual, of a separable H.I. Banach space. Moreover the space of bounded linear operators \({\mathcal{L}}Y\) consists of elements of the form \(\lambda I+W\) where \(W\) is a weakly compact operator and hence it has separable range. Another consequence of the exhibited method is the proof of the complete dichotomy for quotients of H.I. Banach spaces. Namely we show that every separable Banach space \(Z\) not containing an isomorphic copy of \(\ell^1\) is a quotient of a separable H.I. space \(X\). Furthermore the isomorph of \(Z^*\) into \(X^*\), defined by the conjugate operator of the quotient map, is a complemented subspace of \(X^*\). Readership Graduate students and research mathematicians interested in analysis. Table of Contents  Introduction
 General results about H.I. spaces
 Schreier families and repeated averages
 The space \({X= T [G,(\mathcal{S}_{n_j}, {\tfrac {1}{m_j})}_{j},D]}\) and the auxiliary space \({T_{ad}}\)
 The basic inequality
 Special convex combinations in \(X\)
 Rapidly increasing sequences
 Defining \(D\) to obtain a H.I. space \({X_G}\)
 The predual \({(X_G)_*}\) of \({X_G}\) is also H.I.
 The structure of the space of operators \({\mathcal L}(X_G)\)
 Defining \(G\) to obtain a nonseparable H.I. space \({X_G^*}\)
 Complemented embedding of \({\ell^p}, {1\le p < \infty}\), in the duals of H.I. spaces
 Compact families in \(\mathbb{N}\)
 The space \({X_{G}=T[G,(\mathcal{S}_{\xi_j},{\tfrac {1}{m_j})_{j}},D]}\) for an \({\mathcal{S}_{\xi}}\) bounded set \(G\)
 Quotients of H.I. spaces
 Bibliography
