Associate Spaces
In this chapter, we study associate spaces X1 of symmetric spaces X. The space X1 is defined by the duality \(\langle f,g\rangle =\int gf\,dm\) , f ∈ X, g ∈ X1, and the norm \(\|\cdot \|_{\mathbf{X}^{1}}\) is induced by the canonical embedding of X1 into
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Associate Spaces
In this chapter, we study associate Rspaces X1 of symmetric spaces X. The space X1 is defined by the duality hf ; gi D fg dm, f 2 X, g 2 X1 , and the norm k kX1 is induced by the canonical embedding of X1 into the dual space X of X. We show that .X1 ; k kX1 / is a symmetric space and that the canonical embedding of X1 into X is surjective if and only if the space X is separable, i.e., X has property .A/.
7.1 Dual and Associate Spaces Let X be a symmetric space and let X be its dual Banach space. The space X consists of all linear continuous (bounded) functionals u W X ! R on X equipped with the norm kukX D supfju.f /j W kf kX 1g < 1: Every symmetric space X is a Banach ideal lattice. Hence its dual space X is also a Banach ideal lattice with the natural order u v ” u.f / v.f / for all f 2 X; f 0: For every u 2 X , one can define functionals juj, uC , and u such that u D uC u ; juj D uC C u : The functional juj is the least element in X such that juj u juj and juj.f / 0 for all f 2 X, f 0.
© Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_7
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7 Associate Spaces
In some cases, the dual space X of a symmetric space X is itself a symmetric space, or to be more precise, it can be identified in a natural way with a symmetric space. Consider two typical examples. Example 7.1.1. Let X D L1 . Then L1 D L1 by Theorem 2.4.1. More precisely, for every function g 2 L1 , there is ug 2 L1 , defined by Z1 ug .f / D
fgdm; f 2 L1 : 0
Then ug 2 L1 and kug kL1 D kgkL1 . Indeed, if u 2 L1 , the equality u .A/ D u.1A /; A 2 Fm determines a -additive set function u on RC , which is absolutely continuous with respect to the measure m. By the Radon–Nikodym theorem, there exists g such that Z u .A/ D
gdm A
for all A 2 Fm with mA < 1, and hence Z1
Z1 fdu D
u.f / D 0
fgdm D ug .f / 0
for all f 2 L1 .m/. Thus, the embedding W L1 3 g ! ug 2 L1 is an isometric isomorphism between L1 and L1 . Example 7.1.2. Let X D L1 . For every function g 2 L1 , there exists a functional ug 2 L1 , Z1 ug W L1 3 f !
fgdm 2 R; 0
such that the mapping W L1 3 g ! ug 2 L1 is a linear isometry of L1 into L1 , i.e., kug kL1 D kgkL1 . However, in this case, .L1 / D fug ; g 2 L1 g does not coincide with the space L1 . In other words, not all functionals u 2 L1 have the form ug ; g 2 L1 .
7.2 The Maximal Property of Products f g
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Indeed, since clL1 .F0 / D L01 ¤ L1 ; we can choose u1 2 L1 such that u1 .1Œ0;1/ / D 1 and u1 .f / D 0 for all f 2 F0 . Z1 If u1 D ug1 for some g1 2 L1 , then we would have g1 dm D 1 and Z1
0
fg1 dm D 0 for all f 2 F0 , which is false. Thus, the embedding .L1 / L1 is
0
strict (see also Theorem 2.4.1). Considering a symmetric space X, we would like to characterize the part of the space X that consists of all functionals ug 2 X ; g 2 L1 C L1 , while all such functions g form a symmetric
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