The strict topology : on C(S), the bounded continuous complex valued functions on the locally compact Hausdorff space S, was first introduced by R. C. Buck [3], [4], [5]. It has also been studied by I. Glicksberg [10], J. Wells [20], and C. Todd [17]. This topology has been used in the study of various problems in spectral synthesis [11], spaces of bounded holomorphic functions [15], and multipliers of Banach algebras [18], [19]. This paper is a detailed account of results announced by the author in [6], [7] on the relationship of C(S),B with its dual M(S), the bounded Radon measures on S. In particular, we are concerned with the question (posed by Buck) of whether or not C(S),B is a Mackey space and, consequently, with compactness criteria in M(S). The existence and description of the Mackey topology, the strongest topology yielding a given adjoint space, is known, and there are several properties (e.g., metrizable) which imply that a designated topology is the Mackey topology. However, the author knows of no example of a topological vector space with an intrinsically defined topology which is a Mackey space, except by virtue of some formally stronger property (e.g., metrizable, barrelled, bornological). This is not true for C(S),B. In fact, we show that if S is paracompact then every :-weak* countably compact subset of M(S) is /-equicontinuous; consequently, C(S),B is a Mackey space (Theorem 2.6). Also, if S is not compact then C(S),B is not barrelled, bornological, nor metrizable. It can also happen that C(S),B is not a Mackey space, as we show for the case when S is the space of ordinal numbers less than the first uncountable ordinal. In ?3 we examine the subspace problem for C(S),B. That is, if C(S),B is a Mackey space, which subspaces of C(S) are Mackey spaces when furnished with the relative strict topology? We are able to solve this problem when S is the space of positive integers. Also, we show that H O, the bounded holomorphic functions on the open unit disk D, is not a Mackey space when endowed with the : topology-even though C(D), is. From these results we prove the existence of a closed subspace N of 11 such that there is no bounded projection of 11 onto N. Finally, (1V, O) is a semireflexive Mackey space with a closed subspace which is not a Mackey space.