ON COMPLETION OF SPACES OF WEAKLY CONTINUOUS FUNCTIONS J. FERRERA, J. GOMEZ GIL AND J. G. LLAVONA In this paper we find a useful representation of the completion of C W (E, F), the space of weakly continuous functions between two Banach spaces E and F. We introduce the following spaces: C wb {E, F) and C wk {E, F), which contain the functions which are weakly continuous when restricted to bounded sets and to weakly compact sets, respectively. And C WSC (JE, F) which contains the functions which are weakly sequentially continuous. We study relations between these spaces, and we obtain conditions about the Banach space E for which they represent the completion of C W (E,F). Throughout this paper E and F will denote Banach spaces, F ± {0}, C{E, F) the space of continuous functions from E to F. For each n e N, ty{ n E, F) will denote the space of all continuous n-homogeneous F-valued polynomials on E. Finally, if £ and F are complex Banach spaces §(£, F) will denote the spaces of all holomorphic functions between E and F. When the range space is not specifically identified, then it will be assumed that the range is the scalar field. 1. DEFINITION. Let E and F be Banach spaces and A <= E. A function /: A -* F is said to be weakly continuous if for each xe A and £ > 0, there are </>!,..., <t> k e E' and 6 > 0 such that if ye A, Ifaix-y)] < d (i = 1, ...,k) then ||/(x)-/(y)|| < e. We will denote by C w (£, F) the space of all weakly continuous functions from E to F, and by C wb (E, F) (resp. C wk (E, F)) the space of all functions from E to F which are weakly continuous when restricted to bounded sets (resp. weakly compact sets). Clearly we have that C W (E, F) c C wb (E, F) c C wk {E, F). Moreover, C W (E, F) = C wb {E, F) if and only if E is finite dimensional, because if there exists a sequence {</>„} of linearly independent continuous linear forms on E such that ||0J| = 1 for every 00 n e N, then the polynomial Q = £ 2~ n 0* ® y, where y € F, y =£ 0, is not weakly n=l continuous (see [2]); however it is easy to see that Q e C wb (E, F). We consider C W (E, F), C wb {E, F) and C wk (E, F) endowed with the topology of uniform convergence on weakly compact subsets of E. 2. PROPOSITION. C W (£, F) is a dense subspace ofC wk {E, F). Proof. Let / e C wk (E, F). For every K <= E weakly compact, we consider f\ K : (K, a{E, E')\ K ) -» (F, || ||). Since {K, a{E, E')\ K ) is compact (throughout compact means compact and separated), and hence collectwise normal, it follows from a result of Dowker ([3]) that there exists f K \E-*F, a weakly continuous Received 14 July, 1982. Bull. London Math. Soc, 15 (1983), 260-264