5 - L^p Spaces; Introduction to Functional Analysis

Summary

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) such as the unit interval with Lebesgue measure. For any two such functions, f and g, we have a finite integral ∫ f g d μ = ∫ f(x)g(x)d μ(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)^½, 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 ∫ f² 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 d μ < ∞. This collection is called ^p or ^p(μ).