Bounded Linear Functionals and Best Approximation from Hyperplanes and Half-Spaces

Abstract

In this chapter we study the dual space of all bounded linear functionals on an inner product space. Along with the metric projections onto Chebyshev subspaces, these functionals are the most important linear mappings that arise in our work. We saw in the last chapter that every element of the inner product space X naturally generates a bounded linear functional on X (see Theorem 5.18). Here we give a general representation theorem for any bounded linear functional on X (Theorem 6.8). This implies the Fréchet-Riesz representation theorem (Theorem 6.10), which states that in a Hilbert space every bounded linear functional has a “representer.” This fact can be used to show that every Hilbert space is equivalent to its dual space. In general (incomplete) inner product spaces we can characterize the bounded linear functionals that are generated by elements, or representers, in X. They are precisely those that “attain their norm” (Theorem 6.12). We should mention that many of the results of this chapter—particularly those up to Theorem 6.12—can be substantially simplified or omitted entirely if the space X is assumed complete, i.e., if X is a Hilbert space. Because many of the important spaces that arise in practice are not complete, however (e.g., C ₂[a, b]), we have deliberately avoided the simplifying assumption of completeness. This has necessitated developing a somewhat more involved machinery to handle the problems arising in incomplete spaces.