Nonlocally Convex Functional Analysis
In this chapter the goal is to explore the implications of our general metrization theory to aspects of functional analysis in nonlocally convex topological vector spaces. Some of the concrete topics studied here deal with the completeness and separabilit
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Nonlocally Convex Functional Analysis
The goal in this chapter is to establish completeness and separability criteria for large classes of topological vector spaces that are typically nonlocally convex (such as Lebesgue-like spaces, Lorentz spaces, Orlicz spaces, mixed-normed spaces, tent spaces, and discrete Triebel–Lizorkin and Besov spaces) and to derive pointwise convergence results in the case of vector spaces of measurable functions. The proofs of our results in this chapter make essential use of abstract capacitary estimates. By a capacity we will understand a nonnegative function C defined in some algebraic environment G equipped with some associative binary operation that is allowed to be only partially defined (i.e., its domain could, in principle, be just a subset of G G) and that is quasisubadditive. The latter property indicates that there exists a constant c 2 Œ0; C1/ such that C .f g/ c C .f / C C .g/ whenever f; g 2 G have a meaningfully defined product f g 2 G. This topic was the subject of extensive work in earlier chapters. Here we use this analysis and further expand upon it. For example, we will employ capacitary estimates in such settings as the case when .G; / is the underlying Abelian additive group of a given vector space X (in which scenario, C may be allowed to be a quasinorm on X ), when .G; / consists of a sigma-algebra of sets M equipped with the operation of taking unions, or, more generally, when G is a lattice X , with f g taken to be f _ g WD supff; gg for each f; g 2 X (in which case C may be thought of as a rough version of a measure). The presentation in this chapter follows closely [82].
5.1 Formulation of Results Typically, completeness results are proved via ad hoc methods by reducing matters to the completeness of other, more standard spaces (a case in point is the treatment of tent spaces from [33]) or, when done abstractly, such considerations are largely limited to genuine Banach spaces (as is the case with the treatment in Chap. 15 of D. Mitrea et al., Groupoid Metrization Theory, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-8397-9 5, © Springer Science+Business Media New York 2013
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5 Nonlocally Convex Functional Analysis
the classical monograph [130] of Zaanen, or the more timely presentation in Theorem 1.7, p. 6 in the monograph [15] by Bennett and Sharpley). More specifically, in [15, 74, 130] (as well as in many other works based on these references), the authors consider K¨othe function spaces, i.e., having fixed a background measure ˚ space, spaces of the form L WD f measurable W kf k WD .jf j/ < C1 , where is a mapping defined on MC , the collection of all nonnegative measurable functions, satisfying for each f 2 MC ; .f / 2 Œ0; C1; and .f / D 0 , f D 0;
(5.1)
.f / D .f / for each f 2 MC and each 0;
(5.2)
.f C g/ .f / C .g/ for each f; g 2 MC ;
(5.3)
.f / .g/ whenever f; g 2 MC satisfy f g a.e.
(5.4)
In particular, the subadditivity property (5.3) precludes one from considering ar
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