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The Hopf‐Rinow Theorem is false in infinite Dimensions

The Hopf‐Rinow Theorem is false in infinite Dimensions THE HOPF-RINOW THEOREM IS FALSE IN INFINITE DIMENSIONS C. J. ATKIN The purpose of this note is to give a counterexample to what one might call the " weak form" of the Hopf-Rinow theorem in infinite dimensions. The " strong form ", that between any two points of a complete connected C°° Riemannian mani- fold there is a minimising geodesic, is disproved in infinite dimensions by counter- examples of Grossman [4] and McAlpin [5]; the " weak form", that any two points can be joined by some geodesic, has supposedly been an open question (see e.g. Flaschel-Klingenberg [3; p. 59]). It has apparently not been observed that a trifling modification of Grossman's example furnishes a counterexample to this statement too. The only difficulty is in setting up the example so that the absence of a geodesic can actually be proved—the geometrical intuition is merely Grossman's. LEMMA 1. Let L be a separable real or complex Hilbert space of infinite dimension. There exist bounded operators S, T onL such that T is self-adjoint, S skew-adjoint, and TS-ST >0. Proof Represent L as 1 (N), the space of real or complex square-summable sequences indexed by the natural numbers, with standard basis http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

The Hopf‐Rinow Theorem is false in infinite Dimensions

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/7.3.261
Publisher site
See Article on Publisher Site

Abstract

THE HOPF-RINOW THEOREM IS FALSE IN INFINITE DIMENSIONS C. J. ATKIN The purpose of this note is to give a counterexample to what one might call the " weak form" of the Hopf-Rinow theorem in infinite dimensions. The " strong form ", that between any two points of a complete connected C°° Riemannian mani- fold there is a minimising geodesic, is disproved in infinite dimensions by counter- examples of Grossman [4] and McAlpin [5]; the " weak form", that any two points can be joined by some geodesic, has supposedly been an open question (see e.g. Flaschel-Klingenberg [3; p. 59]). It has apparently not been observed that a trifling modification of Grossman's example furnishes a counterexample to this statement too. The only difficulty is in setting up the example so that the absence of a geodesic can actually be proved—the geometrical intuition is merely Grossman's. LEMMA 1. Let L be a separable real or complex Hilbert space of infinite dimension. There exist bounded operators S, T onL such that T is self-adjoint, S skew-adjoint, and TS-ST >0. Proof Represent L as 1 (N), the space of real or complex square-summable sequences indexed by the natural numbers, with standard basis

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Nov 1, 1975

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