6 - Convex Sets and Duality of Normed Spaces

Summary

Functional analysis is concerned with infinite-dimensional linear spaces, such as Banach spaces and Hilbert spaces, which most often consist of functions or equivalence classes of functions. Each Banach space X has a dual space X′ defined as the set of all continuous linear functions from X into the field ℝ or ℂ.

One of the main examples of duality is for L^p spaces. Let (X, S, μ) be a measure space. Let 1 < p < ∞ and 1/p + 1/q = 1. Then it turns out that L^p and L^q are dual to each other via the linear functional f ↦ ∫ f g d μ for f in ^p and g in ^q. For p = q = 2, L² is a Hilbert space, where it was shown previously that any continuous linear form on a Hilbert space H is given by inner product with a fixed element of H (Theorem 5.5.1).

Other than linear subspaces, some of the most natural and frequently applied subsets of a vector space S are the convex subsets C, such that for any x and y in C, and 0 < t < 1, we have tx + (1 – t)y ∈ C. These sets are treated in §§6.2 and 6.6. A function for which the region above its graph is convex is called a convex function. §6.3 deals with convex functions. Convex sets and functions are among the main subjects of modern real analysis.