Monatshefte fiir Mathematik 75, 296--302 (1971) C) by Springer-Verlag 1971 A Normed Linear Space Containing the Schlicht Functions* By J. A. Cima and J. A. Pfaltzgraff, Chapel Hill, North Carolina, USA (Received ~eptember 16, 1970) 1. Introduction In this paper we study a set X of locally schlicht functions on the open unit disk D. The set X is given a real linear space structure which preserves the local scMichtness of functions in X. We use a known growth condition on art f (z) (for / in X) to obtain a norm for X. We study this norm topology on X and compare it with the (relative) com- pact open topology for X. We show that the normalized schlicht functions are a connected subset of X with this topology. In a recent paper [4] H. tIoRNICH introduced and studied a certain Banach space @ of locally schlicht functions. Hornich requires that all functions in ~ have sup I art/'(z) [ (] z I < 1) finite, and therefore ~ does not contain all of the normalized schlicht functions. Using the same linear structure as ~ but a different norm the present authors [3] stu- died a Banach space ~ that contained ~ as a linear subspace. Again, 2 does not contain all of the normalized schlicht functions. In this paper X contains ~ and ~ as linear subspaces, and X contains all nor- realized schlicht functions. 2. The Normed Linear Structure of X Let ~. ~. denote the set of functions ] that are holomorphic and locally schlicht (f (z) ~ 0) in the open unit disk D, and normalized by /(0) = 0, f(0)~ 1. We let ~ denote the sehlieht functions in ~. ~. A family Y__c ~. ~. is called linearly invariant if for each ] e Y the functions * This research was supported by the U. S. Army Research Office - Durham, Grant DA-ARO-D-31-124-G 1151.