Spaces and Banach Spaces

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Abstract

In this work the assumption of quadratic integrability will be replaced by the integrability of |f (x)| p. The analysis of these function classes will shed a particu­ lar light on the real and apparent advantages of the exponent 2; one can also expect that it will provide essential material for an axiomatic study of function spaces. At present I propose above all to gather results about linear operators defined in certain general spaces, no­ tably those that will here be called spaces of type (B)... Function spaces, in particular L p spaces, play a central role in many questions in analysis. The special importance of L p spaces may be said to derive from the fact that they offer a partial but useful generalization of the fundamental L 2 space of square integrable functions. In order of logical simplicity, the space L 1 comes first since it occurs already in the description of functions integrable in the Lebesgue sense. Connected to it via duality is the L ∞ space of bounded functions, whose supremum norm carries over from the more familiar space of continuous functions. Of independent interest is the L 2 space, whose origins are tied up with basic issues in Fourier analysis. The intermediate L p spaces are in this sense an artifice, although of a most inspired and fortuitous kind. That this is the case will be illustrated by results in the next and succeeding chapters. In this chapter we will concentrate on the basic structural facts about the L p spaces. Here part of the theory, in particular the study of their linear functionals, is best formulated in the more general context of Ba­ nach spaces. An incidental benefit of this more abstract viewpoint is that it leads us to the surprising discovery of a finitely additive measure on all subsets, consistent with Lebesgue measure.