Vol. 28, No. 1, 1969

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On the geometry of the unit ball in the space of real annihilating measures

Patrick Robert Ahern

Vol. 28 (1969), No. 1, 1–7

Our purpose is to study the geometry of the unit ball in the space of real measures on the boundary of a finite Riemann surface that annihilate the analytic functions on that surface with continuous boundary values. We show that the number of boundary components of the surface (and hence the topological type) can be determined from the geometry of this unit ball. More precisely, if the surface has k boundary components then this unit ball has 2k 2 “flat faces” of highest possible dimension. We also get some information on extreme points and conclude with an example to show that the linear structure of this unit ball depends not only on the topology of the surface but also on some of its conformal structure.

Mathematical Subject Classification
Primary: 46.30
Secondary: 30.00
Received: 18 December 1967
Published: 1 January 1969
Patrick Robert Ahern
Department of Mathematics
University of Wisconsin
Madison WI
United States