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In this chapter, we discuss selected properties of Banach spaces of type C(K), where K is a linearly ordered compact space, called a compact line for short. In particular, we present Nakhmanson’s theorem stating that if K is a compact line such that C p (K) is a Lindelof space, then K is second-countable. We also discuss the separable complementation property in the context of compact lines.

This chapter deals with K-analytic and analytic spaces C p (X). Some results due to Talagrand, Tkachuk, Velichko and Canela are presented. A remarkable theorem of Christensen stating that a metrizable and separable space X is σ-compact if and only if C p (X) is analytic is proved. We show that the analyticity of C p (X) for any X implies that X is σ-compact (Calbrix’s theorem). We show that C p (X) is K-analytic-framed in ℝ X if and only if C p (X) admits a bounded resolution. We also gather several equivalent conditions for spaces C p (X) to be Lindelof spaces over locally compact groups X.

We deal with complementably universal Banach spaces. Assuming the continuum hypothesis, there exists a complementably universal Banach space of density ℵ1 for the class of Banach spaces with a projectional resolution of the identity. Similar methods produce a universal preimage for the class of Valdivia compacta of weight ℵ1.

This chapter contains classical results about Baire-type conditions (Baire-like, b-Baire-like, CS-barrelled, s-barrelled) on tvs. We include applications to closed graph theorems and C(X) spaces. We also provide the first proof in book form of a remarkable result of Saxon (extending earlier results of Arias de Reyna and Valdivia), that states that, under Martin’s axiom, every lcs containing a dense hyperplane contains a dense non-Baire hyperplane. This part also contains analytic characterizations of certain completely regular Hausdorff spaces X. For example, we show that X is pseudocompact, is Warner bounded, or C c (X) is a (df)-space if and only if for each sequence (μ n ) n in the dual C c (X)′ there exists a sequence (t n ) n ⊂(0,1] such that (t n μ n ) n is weakly bounded, strongly bounded, or equicontinuous, respectively. These characterizations help us produce a (df)-space C c (X) that is not a (DF)-space, solving a basic and long-standing open question.

In this paper we give some properties of the pairwise perfectly normal spaces defined by Lane. In particular we prove that a space (X, P, Q) is pairwise perfectly normal if and only if every P(Q)–closed set is the zero of a P(Q)–l.s.c. and Q(P)–u.s.c. function. Also we characterize the pairwise perfect normality in terms of sequences of semicontinuous functions by means of a result which contains the known Tong's characterization of perfectly normal topological spaces, whose proof we modify by using the technique of binary relations.

In this paper a proper class of barrelled spaces which strictly contains the suprabarrelled spaces is considered. A closed graph theorem and some permanence properties are given. This allows us to prove the necessity of a condition of a theorem of S. A. Saxon and P. P. Narayanaswami by constructing an example of a non-suprabarrelled Baire-like space which is a dense subspace of a Fréchet space and is not an (LF)-space under any strong locally convex topology.

LetM = {μs:s ∈ S} be a family of scalar bounded finitely additive measures defined on a σ-algebra A . The Nikodym–Grothendieck boundedness theorem states that ifMis simply bounded in A thenMis uniformly bounded in A . In this paper we prove that if V = {Ascr;n1, n2,…,np:p, n1, n2…np ∈ N } is an increasing web in A , then there is a strand {An1n2…ni:i ∈ N } such that ifMis simply bounded in one A n1n2…nithenMis uniformly bounded in A (Theorem 3.1). This result is deduced from the fact that if W = {En1n2…np:p, n1, n2,…,np ∈ N } is a linear increasing web inl0∞(X, A ), then there exists a strand {En1n2…ni:i ∈ N } such that everyEn1n2…niis barrelled and dense inl0∞(X, A ) (Theorem 2.7). From this strong barrelledness condition previous results of the author jointly with J. C. Ferrando are improved here. These results are related to the classical result of Diestel and Faires in vector measures.

Let $C_{0}\left( \Omega ,X\right)$ be the linear space of all continuous functions from a locally compact normal space $\Omega $ into a normed space $X$ vanishing at infinity, equipped with the supremum-norm topology. The main result of the paper says that if $X$ is barrelled, then the space $C_{0}\left( \Omega ,X\right) $ is always barrelled. This answers a question posed by J. Horváth.