CHAPTER 2 Preliminaries 2.1. Notations We will use the following standard notations, usually without comment. LpB1,...,Bk B1q is the space of bounded k-linear maps B1 ˆ ¨ ¨ ¨ ˆ Bk Ñ B1. In particular, with B1 R, we have the space of bounded k-linear forms. When B1 ¨ ¨ ¨ Bk B we also write LpBk B1q. denotes the dual space of the Banach space B, i.e., the space LpB Rq of bounded linear functionals B Ñ R. If x P B and P B˚, we use the notations x˚pxq and xx˚, xy, or xx, x˚y, as synonyms. We write K B1, for a subset B1 Ď B, if xx˚, xy 0 for every x P B1. We use several standard Banach spaces in our results and examples for conve- nience we recall their definitions here. For any set S and p P r1, 8q, ppSq is the Banach space of all functions S Ñ R such that the norm }f } ppSq :“ p ř sPS |f psq|pq1{p is finite. (We are mainly interested in the cases p 1, 2.) Further, 8pSq is the Banach space of all bounded functions S Ñ R, with the norm }f } 8 pSq :“ supsPS |f psq|. We use sometimes the notation fs for f psq, in particular when S N. We define the support of f as supppf q :“ ts : f psq 0u, and note that if f P ppSq, with p ă 8, then supppf q is countable even if S is uncountable. c0pSq is the space of all function f : S Ñ R such that ts : |f psq| ą εu is finite for each ε ą 0 this is a closed subspace of 8, and is thus a Banach space with the inherited norm }f }c0 pSq :“ }f } 8 pSq :“ supsPS |f psq|. Note that every element of c0pSq has countable support. es denotes the function esptq :“ 1tt su that is 1 at s and 0 everywhere else (defined for t in some set S, which will be clear from the context). Let c00pSq be the space of all functions f : S Ñ R with supppf q finite this is the linear span of tes : s P Su. Then c00pSq Ď c0pSq Ď 8pSq, and c00pSq is dense in c0pSq. Hence c0pSq is the closed linear span in 8pSq of the functions tes : s P Su. It follows that c00pSq˚ c0pSq˚ 1pSq, with the standard pairing xf, gy ř sPS f psqgpsq. When S N, we write just p and c0. LppSq, where S pS, S,μq is a measure space and p P r1, 8q, is the space of all measurable functions f : S Ñ R such that ş S |f |p ă 8 (as usual identifying functions that are equal a.e.). CpKq, where K is a compact topological space, is the space of all continuous functions f : K Ñ R, with the norm supK |f |. (We are particulary interested in Cr0, 1s.) 5
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