8 - Montel and nuclear spaces

Summary

In this chapter we study the class of the so-called semi-Montel spaces and two important subclasses consisting of the nuclear spaces and the Montel spaces, respectively. We will see that for members of these classes the duality and reflexivity theory becomes more powerful and varied than for arbitrary spaces.

We first study compactoid operators (i.e., operators mapping some zero neighbourhood onto a compactoid set, 8.1.1) and compactifying operators (i.e., continuous operators that map bounded sets onto compactoids, 8.3.1). Compactoid operators are compactifying (8.3.2) but the converse does not hold (8.3.4). Basic properties of compactoid operators are listed in 8.1.3, the general form of a compactoid operator into c 0 is given in 8.1.9(ii). As an application we derive in 8.2.1 and 8.2.2 that, if the valuation of K is dense, there is no continuous linear surjection ℓ^∞ → c₀, and that ℓ^∞ does not have a base. The general form of a compactifying operator into c 0 is given in 8.3.9.

In Section 8.4 we treat semi-Montel spaces E, i.e., for each normed space F each T ∈ L (F, E) is compactoid, 8.4.1(i); equivalently, for each normed space F each T ∈ L (E, F) is compactifying, 8.4.5(ε). It is also proved in 8.4.5(δ) that a space is semi-Montel if and only if each bounded set is a compactoid. For polar spaces (spaces of countable type) E we characterize semi-Montelness in terms of E′_b in 8.4.8 (8.4.13). For this we provide a new characterization of compactoids in polar spaces (8.4.9). Hereditary properties of semi-Montel spaces and reflexive semi-Montel spaces, called Montel spaces (8.4.2), follow in 8.4.24, 8.4.25 and 8.4.26.