Here we concentrate on general properties of random sums that remain valid in arbitrary Banach spaces, while more specific results connected with the geometry of the underlying space are taken up in the following chapter.

## Analysis in Banach Spaces

In the present chapter, we establish a range of comparison results relating the L p norms of different types of random sums, as well as different L p norms of a fixed sum. We also describe the dual and bi-dual of the spaces of random sequences, and characterise the convergence of infinite random series. In the final section, we compare the L p norms of random sums and lacunary trigonometric sums. In this chapter we connect some of the deeper properties of Rademacher sums and Gaussian sums to the geometry of the Banach space in which they live.

### Sitater per år

We begin with a study of the notions of type and cotype, defined through nontrivial upper and lower bounds for random sums. Under the assumption of finite cotype, we prove a refined version of the contraction principle involving function instead of constant coefficients; for this we also develop a necessary minimum of the theory of summing operators. The fourth section is devoted to the notion of K -convexity and its connections with the duality of the random sequence spaces; this section culminates in Pisier's characterisation of K -convexity in terms of non-trivial type.

The final section investigates the properties of multiple random sums involving products of Gaussian or Rademacher variables. We discuss both the general operator-theoretic mechanisms of creating R -bounded families, and concrete sources and applications of R -boundedness in classical analysis.

## Analysis in Banach Spaces

One section is dedicated to the central role of R -boundedness in the theory of Fourier multipliers, and another one to the R -boundedness of integral means and the range of sufficiently smooth operator-valued functions. In the final section, we characterise the situations in which R -boundedness coincides with other types of boundedness. This chapter presents the theory of radonifying operators and explains their use as generalised square functions, which allows the extension of key ideas from classical Littlewood--Paley theory in L p spaces to more general Banach spaces. We also show that the generalised square function space admits canonical extensions of linear operators bounded on the scalar-valued L 2 space, avoiding the intrinsic difficulties of the L p extension problem discussed in Volume I.

Assuming type, cotype or related properties, we even obtain R -bounded extensions of bounded families of Hilbert space operators. In the final section we prove several embedding theorems of classical function spaces into the space of radonifying operators.

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loatutuamorrpecfa.tk The present volume develops the theory of integration in Banach spaces, martingales and UMD spaces, and culminates in a treatment of the Hilbert transform, Littlewood-Paley theory and the vector-valued Mihlin multiplier theorem.