A bounded linear operator on a Banach space which is one-one, dense and 'relatively almost open' must be invertible. INTRODUCTION In Berberian's book [1] it is shown that, on a reflexive Banach space X, a non-invertible operator T c L(X) which is neither a left nor a right zerodivisor must be both a topological left and a topological right zero-divisor in the Banach algebra L(X) ([1, Corollary 57.1 1]). In this note we show that the space X need not be reflexive, and further strengthen the statement. If T: X -* Y is a bounded linear operator between normed spaces write T^: X -* T(X) for its 'truncation': thus T^ is automatically onto, and (0.1) T one-one T^ one-one and (0.2) T bounded below T^ bounded below. We may refer the reader to [1], [2] or [3] for the concepts of "bounded below", "open" and "almost open": we shall call T E L(X, Y) relatively open if T^ is open and call T relatively almost open if T^ is almost open. The second of these concepts can be expressed in terms of the first, via duality; writing T*: Y* X* for the dual or adjoint of T: X Y, we have Theorem 1. If T E L(X, Y) is a bounded linear operator between normed spaces then (1.1) T relatively almost open T* relatively open. Proof. We have, by the definition of "relatively almost open" and [2,(2.3.4)], T relatively almost open -==T^ almost open 4 (T^)* bounded below and, by the definition of "relatively open", T* relatively open < (T* ) open; Received by the editors June 14, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 47A99; Secondary 46B99. (D 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page