A NOTE ON THE SPACES Lp FOR 0 < p < 1

It is shown that there is no Hausdorff vector topology p on the space Lp (where 0 < p < 1) such that the unit ball of Lp is relatively compact for the topology p. It is well known that the space L1 (0, 1) is not a dual Banach space; this follows from the Krein-Milman theorem. It is not even isomorphic to a dual space, by a result due to Gelfand [2] (see Bessaga and Pelczyn'ski [1] and Namioka [6]). An equivalent statement is that there is no Hausdorff locally convex vector topology p on L1 such that the unit ball of L1 is relatively compact for p. In this note we establish a conjecture due to J. H. Shapiro that for 0 < p < 1 there is no Hausdorff vector topology on the space LP(O, 1) such that the unit ball is relatively compact. For the case p = 1, this extends the previous result as we no longer restrict the topology p to be locally convex. Note that for the space 1p, 0 < p < 1, the topology of coordinatewise convergence makes the unit ball compact. Also the topology of uniform convergence on compact subsets of AX, the open unit disc, makes the unit ball of Hp compact for 0 < p < 1 (cf. [4]). We shall suppose throughout that all vector spaces are real, although the extension to the complex case presents no problems. The norm on Lp is defined by llfllp = Jo If(t)IPdt for 0 < p < 1. We shall also need the space L. (0, 1) of essentially bounded functions with the norm IifILo = ess sup If(t)I. We first gather together some general results. PROPOSITION. Let X be a separable complete p-normed space with unit ball U. Suppose that there exists on X a Hausdorff vector topology such that U is relatively compact. Then (i) there is a metrizable vector topology y on X such that U is y-relatively compact; (ii) if V is the y-closure of U, then V is the unit ball of an equivalent p-norm (i.e. V is bounded); Received by the editors March 2, 1975. AMS (MOS) subject classifications (1970). Primary 46A10, 46E30.