By G. H. Hardy
There will be few textbooks of arithmetic as famous as Hardy's natural arithmetic. considering its e-book in 1908, it's been a vintage paintings to which successive generations of budding mathematicians have grew to become at the start in their undergraduate classes. In its pages, Hardy combines the passion of a missionary with the rigor of a purist in his exposition of the elemental principles of the differential and crucial calculus, of the homes of limitless sequence and of different issues concerning the concept of restrict.
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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).
A course of pure mathematics by G. H. Hardy