Martingales (with discrete time) lie at the centre of this book. They are known to have major applications to virtually every corner of probability theory. Our central theme is their applications to the geometry of Banach spaces.

We should emphasize that we do not assume any knowledge about scalar valued martingales. Actually, the beginning of this book gives a self-contained introduction to the basic martingale convergence theorems for which the use of the norm of a vector valued random variable instead of the modulus of a scalar one makes little difference. Only when we consider the ‘boundedness implies convergence’ phenomenon does it start to matter. Indeed, this requires the Banach space B to have the Radon-Nikodym property (RNP). But even at this point, the reader who wishes to concentrate on the scalar case could simply assume that B is finite-dimensional and disregard all the infinite-dimensional technical points. The structure of the proofs remains pertinent if one does so. In fact, it may be good advice for a beginner to do a first reading in this way. One could argue similarly about the property of ‘unconditionality of martingale differences’ (UMD): although perhaps the presence of a Banach space norm is more disturbing there, our reader could assume at first reading that B is a Hilbert space, thus getting rid of a number of technicalities to which one can return later.

A major feature of theUMDproperty is its equivalence to the boundedness of the Hilbert transform (HT). Thus we include a substantial excursion in (Banach space valued) harmonic analysis to explain this.

Actually, connections with harmonic analysis abound in this book, as we include a rather detailed exposition of the boundary behaviour of B-valued harmonic (resp. analytic) functions in connections with the RNP (resp. analytic RNP) of the Banach space B. We introduce the corresponding B-valued Hardy spaces in analogy with their probabilistic counterparts.We are partly motivated by the important role they play in operator theory, when one takes for B the space of bounded operators (or the Schatten p-class) on a Hilbert space.