Let X denote a locally compact Hausdorff space and Cb(X) the algebra of continuous complex-valued functions on X. The main result of this paper is that X is paracompact if and only if Co(X), the subalgebra of Cb(X) consisting of functions which vanish at infinity, has an approximate identity which is a relatively compact subset of Cb(X) for the weak topology of the pairing of Cb(X) with its strict topology dual. Throughout this paper X denotes a locally compact Hausdorff space, Cb(X) denotes the algebra of continuous complex-valued functions on X, equipped with the strict topology ,3 introduced by Buck [1] and studied by numerous others (for example, see [5, 11], and the surveys [2, 14]), and Co(X) denotes the algebra of continuous complex-valued functions which vanish at infinity on X. Buck proved that the d-dual of Cb(X) is Mt(X), the space of bounded tight (inner regular by compact sets) Borel measures on X. By the weak topology on Cb(X), we mean the topology cT(Cb(X), Mt(X)). An approximate identity for Co(X) is a uniformly bounded net (fa)aEA in Co(X) such that IIhf, hil -+ 0 for each h E Co(X). An approximate identity is said to be well-behaved (a WBAI) if the following conditions hold: (1) 0 N. Taylor [12] introduced the notion of WBAI and showed that if X is paracompact, then Co(X) has a WBAI. We shall use two other abbreviations: WCAI, to stand for an approximate identity for Co(X) that is relatively weakly compact in Cb(X), and TBAI, to denote a f-totally bounded (= equicontinuous) approximate identity for Co(X). Collins and Fontenot [4] studied various classes of approximate identities for Co(X), relationships between these classes, and implications for the topological properties of X. Building on earlier work of Collins and Dorroh [3], they showed that Co (X) has a TBAI if and only if X is paracompact. Subsequently, Wheeler [13] showed that paracompactness is also equivalent to existence of a WBAI for Co(X). His argument uses Stone-Cech compactifications and a set-theoretic lemma due to A. Hajnal. Collins and Fontenot also considered WCAI's: they gave an example, the space of countable ordinals, of a space X such that Co(X) has no WCAI and left open the problem of characterizing spaces X such that Co(X) has a WCAI. In this paper we do three things. First, and principally, we show that Co(X) has a WCAI if and only if X is paracompact, and we observe that our argument also Received by the editors November 25, 1985. 1980 Mathemadics S*ject ()saafwation (1985 Revisio). Primary 46E25; Secondary 46J10, 54D18.