The theory of integration developed in Chapter Three enables us to define certain spaces of functions that have remarkable properties and are of enormous importance in analysis as well as in its applications. We have already, in § 7, considered spaces whose points are functions. In § 7, we considered only the uniform norm ‖‖ u [see (7.3)] to define the distance between two functions. The present chapter is concerned with norms that are defined in one way or another from integrals. The most important such norms are defined and studied in § 13. These special norms lead us very naturally to study abstract Banach spaces, to which § 14 is devoted. While we are not concerned with Banach spaces per se, it is an inescapable fact that many results can be proved as easily for all Banach spaces [perhaps with some additional property] as for the special Banach spaces defined in §§ 7 and 13. Our desires both for economy of effort and for clarity of exposition dictate that we treat these results in general Banach spaces. In § 15, we give a strictly computational construction of the conjugate spaces of the function spaces £ p (1 < p < ∞). We have chosen this construction because of its elementary nature and also because we think that manipulation of inequalities is something that every student of analysis should learn. In § 16, we consider Hilbert spaces, which are.£2 spaces looked at abstractly, and also give some concrete examples and illustrations.
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