In this chapter we study locally convex spaces E having an “orthogonal” base e1, e2, … (9.1.1). We first show that for such E, (weak) sequential completeness, quasicompleteness and completeness are equivalent (9.1.6). E may have closed subspaces and quotients without an “orthogonal” base (9.2.5). We characterize bounded and compactoid sets in E in terms of e1, e2, … (9.2.7) and show that compactoids are metrizable (9.2.9(i)). E is semi-Montel if and only if E′b has an “orthogonal” base (9.2.13). We also characterize semi-Montelness (9.2.15) and nuclearity (9.2.16) in terms of properties of the base.

Every infinite-dimensional Fréchet space contains an infinite-dimensional closed subspace with an “orthogonal” base (9.3.5).

Section 9.4 is a stepping stone for the sequel; here we introduce the perfect sequence spaces and the normal topology in the spirit of the classical spaces of Köthe ([144], 6.30). We prove that it is the class of the spaces E with an “orthogonal” base for which E is weakly sequentially complete and E′ is weakly* sequentially complete (9.4.10).

In Section 9.5 we start with an infinite matrix B of nonnegative real numbers and associate to it a perfect sequence space Λ0(B) in a natural way (9.5.2, 9.5.9); we show that the class of these Λ0(B) is precisely the class of all Fréchet spaces with an “orthogonal” base (9.5.12). This fact turns out to be very useful; it enables us in Section 9.6 to translate (semi-)Montelness and nuclearity into concrete properties of the matrix B (9.6.2, 9.6.3).We apply this in Section 9.7 to spaces of analytic functions for which we prove properties that have been postponed in previous chapters (9.7.5).

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.

Locally convex inductive limits over the real and complex field appear in great abundance in many disciplines of classical analysis and its applications, see [29] for a background account. It is our impression that their non-Archimedean counterparts will play a similar role, see the chapter notes for some supporting evidence on this.

Inductive limits have popped in occasionally in previous chapters, often accompanied by promises to discuss some delicate matters later on. Now that we have the necessary locally convex theory available, we are able to develop some framework on inductive limits, thereby keeping our word.

After recalling the definition of an inductive limit we describe the general form of its continuous seminorms (11.1.1) and zero neighbourhoods (11.1.2). By using basically classical methods we derive results on strictness and regularity, e.g. that every strict inductive sequence of Fréchet spaces is regular (11.1.7). This is followed by a typically non-Archimedean discussion (11.1.8) on weak forms of regularity and strictness. Grothendieck's result saying that a Hausdorff (LF)-space is regular if and only if each bounded set is contained in a Banach disk, is translated without too much effort to the non-Archimedean case (11.1.11). We conclude Section 11.1 with a description of the dual of an inductive sequence being a projective system (11.1.13) and with a few examples (11.1.15).

Stability properties are the main concern of Section 11.2. After recalling results obtained previously in the book (11.2.1) we prove in a modified classical way that the Hausdorff property (11.2.2) and metrizability (11.2.8) are not stable under taking inductive limits.

We first follow the “classical” path by developing the notion of barrelledness, a key tool for the theory of reflexivity to be treated in Section 7.4. Indeed, we will prove (7.1.3) that Fréchet spaces are barrelled, by using the Baire Category Theorem.

Nevertheless, for a proper characterization of reflexivity (see 7.4.13) we need a modification of the notion of barrelledness by introducing the wider concept of “polar barrelledness” in 7.1.6. The fact that it is not identical to “ordinary” barrelledness (7.1.10) is a typical non-Archimedean feature. Despite of this, the proofs of the hereditary properties of (polar) barrelledness for quotients (7.1.12(i)), for locally convex direct sums and inductive limits (7.1.13), for products (7.1.15) and completions (7.1.17) are basically “classical”.

In the same spirit we introduce in Section 7.3 the weak star and strong topologies on the dual E′, define reflexivity in 7.4.1 and prove the characterization 7.4.13 of reflexivity. Products and locally convex direct sums of reflexive spaces are reflexive (7.4.23).

However, after this point our results are going to diverge from the classical ones; therefore, we mention a few facts.

For spherically complete K we show in 7.4.19 that every reflexive space is semi-Montel (i.e., each bounded set is a compactoid, see 8.4.1(i) and 8.4.5(δ)), so that reflexive Banach spaces are finite-dimensional. This is in contrast to 7.4.30 showing that every Fréchet space of countable type over a nonspherically complete K is reflexive.

If K is spherically complete, closed subspaces of reflexive Fréchet spaces are reflexive (7.4.27), but if K is not, the reflexive Banach space ℓ∞ contains a closed non-reflexive subspace (7.4.18(i)).

The most important examples of locally convex spaces occurring in this book are listed below, together with indications where their properties can be found. References in brackets concern underlying aspects and definitions. Numbers refer to theorems, corollaries, etc. unless indicated.