4 - Dual Spaces

Summary

4.1 The Duals of L^p Spaces

Given a normed linear space E, we have briefly encountered the dual space E*, consisting of bounded linear functionals on E. We know that it is a Banach space (Theorem 2.3.13), but beyond that we do not know much. In the finite dimensional case, E and E* are isomorphic as vector spaces. However, in the infinite dimensional case, this need not be true. Moreover, even in the finite dimensional case, an arbitrary isomorphism may not be isometric!

Now, if E is a Hilbert space, then there is an isometric isomorphism E ≅ E* by the Riesz Representation Theorem. This is not true for an arbitrary Banach space. However, the core idea of that proof (specifically, the construction of the map Δ) is applicable when studying the dual space of ℓ^p or L^p[a, b] for 1 ≤ p ≤ ∞. This is the focus of this section, where we explicitly determine these dual spaces, giving new meaning to Hölder's Inequality and revisiting some beautiful measure theory along the way.