eBook ISBN:  9781470404079 
Product Code:  MEMO/170/806.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $39.00 
eBook ISBN:  9781470404079 
Product Code:  MEMO/170/806.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $39.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 170; 2004; 114 ppMSC: Primary 46;
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 \(X^*\).
ReadershipGraduate students and research mathematicians interested in analysis.

Table of Contents

Chapters

Introduction

1. General results about H.I. spaces

2. Schreier families and repeated averages

3. The space $X = T[G, (\mathcal {S}_{n_j}, 1/m_j)_j, D]$ and the auxiliary space $T_{ad}$

4. The basic inequality

5. Special convex combinations in $X$

6. Rapidly increasing sequences

7. Defining $D$ to obtain a H.I. space $X_G$

8. The predual $(X_G)_*$ of $X_G$ is also H.I.

9. The structure of the space of operators $\mathcal {L}(X_G)$

10. Defining $G$ to obtain a nonseparable H.I. space $X^*_G$

11. Complemented embedding of $l^p$, $1 \leq p < \infty $, in the duals of H.I. spaces

12. Compact families in $\mathbb {N}$

13. The space $X_G = T[G, (\mathcal {S}_\xi , 1/m_j)_j, D]$ for an $\mathcal {S}_\xi $ bounded set $G$

14. Quotients of H.I. spaces


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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 \(X^*\).
Graduate students and research mathematicians interested in analysis.

Chapters

Introduction

1. General results about H.I. spaces

2. Schreier families and repeated averages

3. The space $X = T[G, (\mathcal {S}_{n_j}, 1/m_j)_j, D]$ and the auxiliary space $T_{ad}$

4. The basic inequality

5. Special convex combinations in $X$

6. Rapidly increasing sequences

7. Defining $D$ to obtain a H.I. space $X_G$

8. The predual $(X_G)_*$ of $X_G$ is also H.I.

9. The structure of the space of operators $\mathcal {L}(X_G)$

10. Defining $G$ to obtain a nonseparable H.I. space $X^*_G$

11. Complemented embedding of $l^p$, $1 \leq p < \infty $, in the duals of H.I. spaces

12. Compact families in $\mathbb {N}$

13. The space $X_G = T[G, (\mathcal {S}_\xi , 1/m_j)_j, D]$ for an $\mathcal {S}_\xi $ bounded set $G$

14. Quotients of H.I. spaces