REDUCED POWERS OF THE REAL NUMBER SYSTEM AND EQUIVALENTS OF THE HAHN-BANACH EXTENSION THEOREM.
CALIFORNIA INST OF TECH PASADENA DEPT OF MATHEMATICS
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One of the main principles of functional analysis is the so-called Hahn-Banach extension theorem. In the present paper we are mainly interested in the question which other statements in mathematics can be shown to be effectively equivalent to the Hahn-Banach extension theorem. The main result can be phrased as follows The Hahn-Banach extension theorem is effectively equivalent to the statement every non-degenerate Boolean algebra admits a non-trivial measure. The method being used is to consider the linear ring of finite elements in a reduced power of the real number system. It is shown that this linear ring is an abstract M-space in the sense of Kakutani. Some of the consequences of this fact are given in the present paper.
- Theoretical Mathematics