As a corollary of the main resu!t of this paper (Theorem i) we show that if ~ i~ not S-dense, then I~[ ~< 1~ s [. This contains as special cases the results of Sauer [4] and Perles and Shelab [5], and of Karpovski and Milman [3]. Before stating Theorem l, we need one more definition; ~ c 3~ is monotone if f ¢ ~ , g e 3 ~ and f ~ g imply g ¢ ~ . Note ~.hat ~ s is monotone and that a monotone fam...

This paper extends Krein-Rutman's theorem on linear operators leaving an invariant cone in infinite-dimensional Banach spaces to multivalued convex functions. The result is applied to obtain necessary and sufficient conditions for global controllability and reachability of nonlinear discrete-time systems described by convex processes.

We extend recent results by Pisier on K-subcouples, i.e. subcouples of an interpolation couple that preserve the K-functional (up to constants) and corresponding notions for quotient couples. Examples include interpolation (in the pointwise sense) and a reinter-pretation of the Adamyan-Arov-Krein theorem for Hankel operators.

In practice the hypothesis that x(uv) # ~(u’v) for some conjugate U’ of u is almost always satisfied. When x is faithful and u = v-r, then this just asserts that u does not belong to the center of G (though it is an easy exercise to give a direct proof of the theorem in this case). We shall derive Theorem 1 from a general result (Theorem 2) about common constituents of permutation characters. T...

In 1934, Whitney posed the problem of how to recognize whether a function f defined on a closed subset X of R is the restriction of a function of class C. Whitney himself solved the one-dimensional case (i.e., for n = 1) in terms of finite differences [W1, W2, W3], giving the classical Whitney’s extension theorem. A geometrical solution for the case C(R) was given by G. Glaeser [G], who introdu...

In recent years there has been interest in a theorem on positive definite matrices known as Lyapunov's theorem. Several authors have proved generalizations of this theorem, (WIELANDT [29J, TAUSSKY [24J, [25J, [26J , OSTROWSKISCHNEIDER [20J, GIVENS [10J, CARLSON-SCHNEIDER [3J, CARLSON [4J) . Lyapunov's theorem and its generalizations have become known as inertia theorems. In this note we shall u...