By L. Auslander, R. Tolimieri
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Extra info for Abelian Harmonic Analysis, Theta Functions and Functional Analysis on a Nilmanifold
For every rectifiable path defined on Œa; b is increasing and continuous. t the path W Œa; b ! X, the map t 7! L. t / Proof. 12). Let us show that it is continuous. We fix a real number > 0. 8 (length is invariant under change of parameter) and the uniform continuity of , we can find a positive real number Á satisfying the following two properties: of Œa; b satisfying j j Ä Á, we have • For every subdivision L. / V . 2) Now let u and v be two points in Œa; b satisfying 0 Ä u v Ä Á and let be a subdivision of Œa; b satisfying j j Ä Á and \Œu; v D fu; vg.
X be a rectifiable path. We say that is parametrized by arclength if for all u and v satisfying a Ä u Ä v Ä b, we have v u D L. j Œu;v /. In particular, if a path L. / D b a. W Œa; b ! 5. Let W Œa; b ! X be a path parametrized by arclength. Then the map t 7! L. t /, defined on the interval Œa; b, is strictly increasing. Proof. 4. 6. Let W Œa; b ! X be a rectifiable path. Then the path W Œ0; L. / ! 2 is parametrized by arclength. Proof. 3, for all u and v satisfying 0 Ä u Ä v Ä L. /, we have v u D L.
We can use the fact (which we prove in Chapter 5 below) that the projection map on the closed ball Bx is 1-Lipschitz to show that the length of any path joining x and y in En n B is bounded below by the length of a path joining x and y and whose image is contained in the sphere S D @B. Since the length of such a path is bounded below by r, we have, for any path W Œa; b ! En n B joining x and y, L. / r, whereas jx yj D 2r. This shows that En n B is not a length space. The same result holds for En with a closed ball removed.
Abelian Harmonic Analysis, Theta Functions and Functional Analysis on a Nilmanifold by L. Auslander, R. Tolimieri