Duals

This chapter presents some properties of the duals of Lebesgue space L p (Ω;E), when E is a Banach space. It proves Clarkson's theorem and Riesz representation theorems, and states some properties of the canonical injection. If p is finite and E is a Hilbert space, then every linear form on L p (Ω;E) can be represented by an element of L p' (Ω;E'). The chapter shows that such a linear form can also be represented by an element of L p' (Ω;E), which is the content of the Riesz-Frechet representation theorem. It defines the semi-norms that generate the topology on L p (Ω;E)-weak.