By Jorge Alberto Barroso
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Extra resources for Aspects of Mathematics and Its Applications
The subsets of f€(X) containing some set V(K, e) are the neighbourhoods of 0 for a topology on f€(X), called the topology of uniform convergence on compact subsets of X (or the compact-open topology), and for this topology f€ (X) is a locally convex Hausdorff space. Nachbin  proves the following results: (1). f€(X) is barrelled if and only if for every non-compact closed subset F of X there exists a function c/J E f€(X) which is unbounded on F. (2). ce(X) is bornological if and only if X is replete.
Thus Bourbaki's problem amounts to ask for which locally convex Hausdorff spaces E is every u(E', E)-bounded subset of E' equicontinuous. An absolutely convex subset A of a vector space E is said to be absorbing if for every x E E there exists a A E K such that x E AA. A subset T of a topological vector space is said to be a barrel if it is absolutely convex, absorbing and closed. Observe that in a locally convex topological vector space every neighbourhood of 0 contains a neighbourhood which is a barrel.
If "fI is the collection of characteristic functions of compact subsets of X, then C6'''fIb(X) = C6'''fIo(X) is the space C6'(X) with the compact-open topology. (2). If "fI is the set of all positive constants, then C6'''fIb (X) = C6'b (X), C6'''fIo(X) = C6'o(X) with their normed topologies. (3). If "fI is the set of all positive, bounded, upper semicontinuous functions which tend to 0 at infinity, then C6'''fIb(X) is C6'b(X) with the so-called strict topology. (4). If X is locally compact and the union of a countable family of its compact subsets, and if "fI consists of all positive continuous functions, then C6'''fIb(X) = C6'''fIo(X ) is the space C6'c (X) of all continuous functions with compact support, equipped with the 'locally convex inductive limit' topology for which the dual of C6'c (X) is the space of Radon measures on X.