ABSTRACT
The key idea of functional analysis is to consider functions as “points” in a space of functions. To begin with, consider bounded, measurable functions on a finite measure space ( X , S , μ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351076197/b707e6d7-4cb3-4b50-a24d-13dd813ac7b0/content/inline5_116_1.tif"/> such as the unit interval with Lebesgue measure. For any two such functions, f and g, we have a finite integral ∫fg dµ = ∫ f(x)g{x) dµ(x). If we consider functions as vectors, then this integral has the properties of an inner product or dot product (f, g): it is nonnegative when f = g, symmetric in the sense that (f, g) ≡ (g, f), and linear in f for fixed g. Using this inner product, one can develop an analogue of Euclidean geometry in a space of functions, with a distance d(f, g) = (f − g, f − g)112, just as in a finite-dimensional vector space. In fact, if µ is counting measure on a finite set with k elements, (f, g) becomes the usual inner product of vectors in ℝ k. But if µ is Lebesgue measure on [0, 1], for example, then for the metric space of functions with distance d to be complete, we will need to include some unbounded functions f such that ∫f 2 dµ < ∞. Along the same lines, for each p > 0 and µ there is the collection of functions f which are measurable and for which ∫ |f| p dµ < ∞. This collection is called ℒ p or ℒ p (µ). It is not immediately clear that if f and g are in ℒ p , then so is f + g; this will follow from an inequality for ∫ |f + g|p dµ in terms of the corresponding integrals for f and g separately, which will be treated in §5.1. In §§5.3 and 5.4, the inner product idea, corresponding to p = 2, is further developed.
