Download Differential Geometry, Part 2 by Chern S., Osserman R. (eds.) PDF

By Chern S., Osserman R. (eds.)

ISBN-10: 082180247X

ISBN-13: 9780821802472

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Additional resources for Differential Geometry, Part 2

Example text

If # is a measure and M is an Orlicz function, the space LM(/Z) is the collection of all/z-measurable functions f for which there exists C > 0 so that f M ( f / C ) d / z < ~ . Then IIf IIM is defined to be the infimum of those C > 0 for which f M ( f / C ) d/z < 1. This is a norm on LM(/z) which makes LM(/z) into a Banach lattice. I f / z is counting measure on N, LM(/z) is called an Orlicz sequence space and is denoted by gM. If 1 ~< p < ~ and W is a positive nonincreasing continuous function on (0, ~ ) so that W(t) --+ ~ as t ~ 0 + , W(t) --+ 0 as t --+ c~, f l W(t) dt -- 1, and f o W(t) dt -- cx~, the Lorentz space LW, p(/z) is the space of all/z measurable functions f for which IlflIw, p "-(f~xz f , ( t ) P W ( t ) d t ) l / p ' where f * is the decreasing rearrangement of Ifl.

We shall discuss that aspect and other equivalences of the RNP later in this section. e. derivative f l , if it exists, is also separably valued and thus is a measurable function since x * f is measurable for each x* in X*. First let us get a feeling for which spaces have the RNP. It is clear from the definition that a subspace of a space with the RNP has the RNP and that a space all of whose separable subspaces have the RNP has the RNP as well. The mapping t w-~ l(0,t) from (0, 1) into L 1(0, 1) shows that L 1(0, 1) fails the RNP.

5] or write down a proof for Lp(lZ ) and see that the H61der type inequalities (4), (5) allow a translation to the lattice setting). A Banach lattice X is called p-convex (p-concave) if the identity operator Ix on X is pconvex (p-concave) and we then define M(P)(X) := M(P)(Ix) and M(p)(X) :-- M(p)(Ix). These constants are called the p-convexity and p-concavity constants of X. So X is pconvex (p-concave) if and only if X* is p*-concave (p*-convex). 8]. In particular, a lattice which is both p-convex and p-concave is lattice isomorphic to an abstract Lp space.

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Differential Geometry, Part 2 by Chern S., Osserman R. (eds.)


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