ABSTRACT

In the previous chapters we have considered function spaces where the values of the functions are in Hilbert spaces or Banach spaces. In this chapter we continue this focus, but here we will take the underlying function spaces to be discrete Banach function spaces. Thus it will be natural to consider our spaces to be sums of Banach spaces with certain properties. Such sums can be realized as special cases of what M.M. Day called substitution spaces. These are defined as follows. Suppose S is an index set and E is a Banach space of scalar functions with the property that if y ∈ E and |x(s)| ≤ |y(s)| for all s ∈ S, then x ∈ E and ‖x‖ ≤ ‖y‖. For each s suppose that Xs is a Banach space and let X = E((Xs)s∈S) denote the functions x such that x(s) ∈ Xs for each s ∈ S and the function |x| ∈ E where |x|(s) = ‖x(s)‖. Then X is a Banach space with the norm ‖x‖ = ‖ |x| ‖E . If each Xs is equal to the same space Y , we write X = E(Y ). The condition put on E above means that it must have what is usually called an absolute norm; that is, the norm of a function depends only on its absolute values.