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We show that the fact that X has a compact resolution swallowing the compact sets characterizes those C c .X/ spaces which have the so-called G-base. So, if X has a compact resolution which swallows all compact sets, then C c .X / belongs to the class G of Cascales and Orihuela (a large class of locally convex spaces which includes the (LM) and (DF)-spaces) for which all precompact sets are metrizable and, conversely, if C c .X/ belongs to the class G and X satisfies an additional mild condition, then X has a compact resolution which swallows all compact sets. This fully applicable result extends the classification of locally convex properties (due to Nachbin, Shirota, Warner and others) of the space C c .X / in terms of topological properties of X and leads to a nice theorem of Cascales and Orihuela stating that for X containing a dense subspace with a compact resolution, every compact set in C c .X / is metrizable.In what follows, unless otherwise stated, X will be a Hausdorff completely regular space and C p .X/ and C c .X / will denote the space C.X/ of all real-valued continuous functions defined on X provided with the pointwise convergence topology and with the compact-open topology, respectively.Let us recall that a family A D ¹A˛W˛2 N N º of subsets of a set X is called a resolution of X if S ¹A˛W˛2 N N º D X and A˛Â Aˇfor˛Ä( coordinatewise), see [9, Chapter 3]. A resolution A of a topological space X is called compact if it consists of compact sets.A locally convex space (lcs) E is said to have a G-base (or a G-basis) (see [9, Chapter 1]) if there exists a basis ¹U˛W˛2 N N º of (absolutely convex) neighborhoods of the origin in E, such that UˇÂ U˛whenever˛Äˇ.An lcs E is said to belong to the class G if its topological dual E 0 has a resolution ¹A˛W˛2 N N º such that for every˛2 N N each sequence in A˛is

Journal of Mathematical Analysis and Applications 324 (2006) 862-881. doi:10.1016/j.jmaa.2005.12.059Received by publisher: 2005-04-22Harvest Date: 2016-01-04 12:20:16DOI: 10.1016/j.jmaa.2005.12.059Page Range: 862-88

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