* Preliminaries* Numbers* Sequences and series* Limits and continuity* Differentiation* The elementary transcendental functions* Integration* Infinite series and infinite products* Trigonometric series* Bibliography* Other works by the author* Index

An introduction to classical real analysis, by Karl R. Stromberg. Pp 575. $29·95. 1981. ISBN 0-534-98012-0 (Wadsworth)

A Banach space X is said to be Hereditarily Indecomposable (HI) if for any pair of closed subspaces Y , Z of X with Y ∩ Z = {0}, Y + Z is not a closed subspace (Throughout this section by the term “subspace” we mean a closed infinite-dimensional subspace of X ) The HI spaces form a new and, as we believe, fundamental class of Banach spaces The celebrated example of a Banach space with no unconditional basic sequence, due to W Gowers and B Maurey ([GM]), is the first construction of a HI space It is easily seen that every HI space does not contain any unconditional basic sequence Actually, the concept of HI spaces came after W Johnson’s observation that this was a property of the Gowers-Maurey example To describe even further the peculiar structure of a HI space, we recall an alternative definition of such a space A Banach space X is a HI space if and only if for every pair of subspaces Y , Z and e > 0 there exist y ∈ Y , z ∈ Z with ||y|| = ||z|| = 1 and ||y − z|| < e Thus, HI spaces are structurally irrelevant to classical Banach spaces, in particular to Hilbert spaces Other constructions of HI spaces already exist We mention Argyros and Deliyanni’s construction of HI spaces which are asymptotic ` spaces ([AD2]), V Ferenczi’s example of a uniformly convex HI space ([F2]) and HI modified asymptotic ` spaces contained in [ADKM] Other examples of Banach spaces which are HI or which have a HI subspace are given in [G1], [H], [OS1] The construction of such a space requires several steps and it uses two fundamental ideas The first is Tsirelson’s recursive definition of saturated norms ([Ts]) and the second is Maurey-Rosenthal’s construction of weakly null sequences without unconditional basic subsequences ([MR]) An important ingredient in the GowersMaurey construction is the Schlumprecht space This is a Tsirelson type Banach space which is arbitrarily distortable and has been used in the solution of important problems Thus beyond its use in the constructions of HI spaces it plays a central role in the solution of the distortion problem for Hilbert spaces ([OS]) The essential difference between Schlumprecht and Tsirelson spaces became more transparent in [AD2] where the mixed Tsirelson spaces were introduced It is natural to expect that HI spaces share special and interesting properties not found in the previously known Banach spaces Indeed, the following theorem is proven in

/pdf/interpolating-hereditarily-indecomposable-banach-spaces-3yr6tf31tz.pdf

Interpolating hereditarily indecomposable Banach spaces

We study some questions concerning the structure of the set of spreading models of a sep- arable infinite-dimensional Banach space X. In particular we give an example of a reflexive X so that all spreading models of X contain l1 but none of them is isomorphic to l1. We also prove that for any countable set C of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of C. In certain cases this ensures that X admits, for each α < ω1, a spreading model (˜ x (α) i )i such that if α < β then (˜ x (α) i )i is dom- inated by (and not equivalent to) (˜ x (β) i )i. Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map.

On the Structure of the Spreading Models of a Banach Space

It is shown that every Banach space either contains $\ell ^1$ or it has an infinite dimensional closed subspace which is a quotient of a H.I. Banach space.Further on, $L^p(\lambda )$, $1<p<\infty $, is a quotient of a H.I Banach space.

We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space $X$. In particular we give an example of a reflexive $X$ so that all spreading models of $X$ contain $\ell_1$ but none of them is isomorphic to $\ell_1$. We also prove that for any countable set $C$ of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of $C$. In certain cases this ensures that $X$ admits, for each $\alpha < \omega_1$, a spreading model $(\tilde x_i^\alpha)_i$ such that if $\alpha < \beta$ then $(\tilde x_i^\alpha)_i$ is dominated by (and not equivalent to) $(\tilde x_i^\beta)_i$. Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map.

On the structure of the spreading models of a Banach space

It is proved that every normalized weakly null sequence has a subsequence which is convexly unconditional. Further, a hierarchy of summability methods is introduced and with this we give a complete classification of the complexity of weakly null sequences.

Convex unconditionality and summability of weakly null sequences

It is proved that every normalized weakly null \sq\ has a sub\sq\ which is convexly unconditional. Further, an Hierarchy of summability methods is introduced and with this we give a complete classification of the complexity of weakly null \sq s.

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Convex unconditionality and summability of weakly null sequences

Convex unconditionality and summability of weakly null sequences