The purpose of this paper is to study the lattice of invariant subspaces of a linear transformation on a finite-dimensional vector space over an arbitrary field. Among the topics discussed are structure theorems for such lattices, implications between linear-algebraic properties and lattice-theoretic properties, nilpotent transformations, and the conditions for the isomorphism of two such lattices. These topics correspond roughly to §§2, 3, 4, and 5 respectively.

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ABSTRACT. The closed convex hull and extreme points are obtained for the starlike functions of order a and for the convex functions of order a. More generally, this is determined for functions which are also Mold symmetric. Integral representations are given for the hulls of these and other families in terms of probability measures on suitable sets. These results are used to solve extremal problems. For example, the upper bounds are determined for the coefficients of a function subordinate to or majorized by some function which is starlike of order a. Also, the lower bound on Re(/(z)/z} is found for each z (\z\ < 1) where/varies over the convex functions of order a and a > 0.Introduction. In this paper we determine the closed convex hulls and extreme points of families of functions which are generalizations of the starlike and convex mappings. These results allow us to solve a number of extremal problems over related families of analytic functions.Let A denote the unit disk {z G C: \z\ < 1) and let A denote the set of functions analytic in A. Then A is a locally convex linear topological space with respect to the topology given by uniform convergence on compact subsets of A. Let S be the subset of A consisting of the functions/ that are univalent in A and

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