L ∞ /c 0 has no Equivalent Strictly Convex Norm

Abstract

It is shown that the quotient space lI/co does not admit an equivalent strictly convex norm. Introduction. We say that a normed space X, liii is strictly convex provided lix + yII 0. Then there is some M E Pa(L) such that Ix*(x)I e-1 and disjoint members Mi (1 ,e. Consider now the vector x = x(l) + x(2) + * +x(d). Obviously lxii = 1 and x*(x) > d. This is the required contradiction. We are now ready to prove the following theorem. THEOREM. Let III III be an equivalent norm on I ?/co. Then III is not strictly convex. PROOF. Let Y = I /co, and S: 1X Y the quotient map be given. Let (en) be a sequence of positive numbers converging to 0. We make the following construction: Take F1 = {x E 1X; lxiil s5 E1 and Received by the editors December 21, 1978. AMS (MOS) subject classifications (1970). Primary 46B05. ? 1980 American Mathematical Society 0002-9939/80/0000-0065/$01 .50