Preprint 49/2001

Generalized lattices in topological vector spaces

Valerii N. Berestovskii, Victor Gichev, and Conrad Plaut

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Submission date: 07. Aug. 2001
Pages: 35
MSC-Numbers: 52C07
Keywords and phrases: lattice, banach space, quotient, fundamental set, universal cover

We study the geometry and topology of generalized lattices in topological vector spaces, and their quotients. A subgroup G of a topological vector space V is a generalized lattice if G is line-free and exhaustive in the sense that the closure of its linear span in V is V. We first discuss stronger conditions than line-free and exhaustive, and give examples of generalized lattices. We show that the sum of a locally generated, exhaustive subgroup G and a cocompact subgroup H in a topological vector space V is dense in V. We next study the groups tex2html_wrap_inline92 for tex2html_wrap_inline94 mainly from the standpoint of geometry. We define geometrically significant global and local contractions of tex2html_wrap_inline96 and tex2html_wrap_inline98, and prove self-similarity properties of tex2html_wrap_inline96. We next study the rectifiability and differentiability properties of curves in Banach spaces and tex2html_wrap_inline96, especially the singular nature of curves in tex2html_wrap_inline96. Following this we study when the (generalized) universal cover of a quotient V/G is again a topological vector space. Finally, we prove the existence of a (possibly non-convex) Dirichlet-Voronoi fundamental set of G in V, when V is a uniformly strongly convex Banach space and V/G is geodesic space. We also show this existence of such a fundamental set for the subgroup tex2html_wrap_inline96 of tex2html_wrap_inline118. We conclude with unsolved questions.

24.11.2021, 02:11