By Victor L. (editor) Klee
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Additional info for Convexity (Proceedings of symposia in pure mathematics, Vol.7)
Since is a positively homogeneous norm-preserving honieomorphism of onto P1o,4-d, it is easy to verify that 'j is a positively homogeneous norm-preserving biunique transformation of L fl PB onto L. We wish to extend i, to the entire positive cone PB (which is the closure of L fl PB) and then to prove that the extended is a homeomorphism of PB onto E. The following fact is crucial: 10. Suppose XE PB and Then the sequence x,ô. for each n €1. is a Cauchy sequence. , — 'ix" I = 0 and ('iX'),E, must be a Cauchy II Three of the above steps are immediate from the relevant definisequence.
Ilane H contains M. — M onto J which carries H — M onto H. Let u denote the natural homeomorphism of F! onto its quotient Since HIM is an infinite-dimensional normed linear space, a theorem in [19) guarantees the existence in HIM of a sequence C, of unPROOF. space HIM. bounded but linearly bounded closed convex bodies such that 0€ mt C,, always mt C,, C,,+1, and nrc,, = 0. For each n let p,, be a point which is interior to the set tr'C, relative to H. Let z E J — H and for each n let B,, u'C,, + [—1/(n + 1), l/(n + 1)]z.
1,2, and 3 fail (or their inverses fail) Many of the homeomorphisms of to be uniformly continuous. The results of § 4 show that this is not accidental; in particular, a bounded convex subset of a normed linear space does not admit a uniformly continuous mapping onto any unbounded metric space. § 5 discusses some unsolved problems and indicates still another way of proving that Hubert space is homeomorphic with its unit cell. 1. A reduction theorem. The homeomorphisms which we seek are obtained from the composition of several simple transformations.
Convexity (Proceedings of symposia in pure mathematics, Vol.7) by Victor L. (editor) Klee