By W.B. Johnson, J. Lindenstrauss

The instruction manual offers an summary of so much points of contemporary Banach area conception and its purposes. The updated surveys, authored by way of prime learn staff within the quarter, are written to be available to a large viewers. as well as featuring the cutting-edge of Banach area idea, the surveys talk about the relation of the topic with such parts as harmonic research, complicated research, classical convexity, chance concept, operator idea, combinatorics, good judgment, geometric degree conception, and partial differential equations.The guide starts off with a bankruptcy on easy techniques in Banach house idea which includes all of the heritage wanted for interpreting the other bankruptcy within the instruction manual. all of the twenty one articles during this quantity after the elemental options bankruptcy is dedicated to at least one particular path of Banach house thought or its functions. each one article includes a stimulated advent in addition to an exposition of the most effects, equipment, and open difficulties in its particular path. so much have an in depth bibliography. Many articles comprise new proofs of identified effects in addition to expositions of proofs that are not easy to find within the literature or are just defined within the unique study papers.As good as being important to skilled researchers in Banach house concept, the instruction manual will be an exceptional resource for proposal and data to graduate scholars and starting researchers. The guide should be priceless for mathematicians who are looking to get an idea of a few of the advancements in Banach

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**Example text**

Now suppose that v is a finite separable measure and r is an X valued measure which is absolutely continuous with respect to v. Then # := Irl is a finite scalar measure which is absolutely continuous with respect to v, so by the scalar Radon-Nikod3)m theorem there is a v-measurable function f >~ 0 so that #(A) = fA f dv for every v-measurable set A. Of course, IX is then also a separable measure and, as we have already remarked, r satisfies (14), so from what we already have proved there is an X valued Ix-measurable function g so that r(A) = f A g dix for every Ix-measurable set A.

If X satisfies this condition only for all separable finite scalar measures, we say that the separable Radon-Nikodym theorem holds in X (a measure # is called separable provided L1 (#) is separable). The usual definition is that a Banach space X has the RNP provided the Radon-Nikodym theorem holds in X and this is equivalent to saying that the separable Radon-Nikod3~m theorem holds in X (see [8, Chapter III]). Later we prove this equivalence for separable X, but first we show a general space X has the RNP if and only if the separable Radon-NikodSm theorem holds in X.

Xn, the expression f ( x l , . . , xn) can be d e f n e d in such a way that any lattice inequality that is true in ]~n is true in X; that is, if g is another continuous homogeneous function on ]1~n and f (a) ~< g (a) for all a = (ot l . . . an) in It~n , then f (x I . . . Xn) <~ g (x 1. . . Xn) for all xl . . . Xn in X. For 1 ~< p ~< cx~, a linear mapping T from a Banach space into a Banach lattice is called p-convex if there exists a constant M so that for all finite sets of vectors in the domain space the following inequality holds: 1/p [rxnl p n=l <~ M Ilxnll p n=l (7) Basic concepts in the geometry of Banach spaces 27 The smallest such M is denoted by M(P)(T).

### Handbook of the Geometry of Banach Spaces: Volume 1 by W.B. Johnson, J. Lindenstrauss

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