Preview of vector-valued integrals

In contrast to construction of integrals as limits of Riemann sums, the Gelfand-Pettis characterization is a property no reasonable notion of integral would lack. Since this property is an irreducible minimum, this definition of integral is called a weak integral. Uniqueness of the integral is immediate when the dual V ∗ separates points, meaning that for v 6 v′ in V there is λ ∈ V ∗ with λv 6= λv′. This separation property certainly holds for Hilbert spaces: the map λw = 〈w, v− v′〉 is a continuous linear functional and λ(v− v′) 6= 0 gives λv 6= λv′. The separation property for Banach spaces is part of the Hahn-Banach theorem. [2] [3] For the rest of this discussion, all topological vector spaces are assumed locally convex without further mention.