Polar lattices from the point of view of nuclear spaces

Sin resume

A Beck—Fiala-type Theorem for Euclidean Norms

Let D be an ellipsoid in ℝn with centre at 0 and principal semi-axes λ1,..., λn. Let u1,..., um ∈ D. It is proved that there exist signs θ1,..., θm = ± 1 such that ∥∑i=1mθiui∥≤λ:=(∑j=1nλj2)12. Furthermore, to each k = 1,..., m there corresponds a subset I of {1,..., m} consisting of exactly k elements, such that ∥∑i∈Iui−(k/m)∑i=1mui∥≤λ

Summable families in nuclear groups

Nuclear groups form a class of abelian topological groups which contains LCA groups and nuclear locally convex spaces, and is closed with respect to certain natural operations. In nuclear locally convex spaces, weakly summable families are strongly summable, and strongly summable are absolutely summable. It is shown that these theorems can be generalized in a natural way to nuclear groups

Additive subgroups of topological vector spaces

The Pontryagin-van Kampen duality theorem and the Bochner theorem on positive-definite functions are known to be true for certain abelian topological groups that are not locally compact. The book sets out to present in a systematic way the existing material. It is based on the original notion of a nuclear group, which includes LCA groups and nuclear locally convex spaces together with their additive subgroups, quotient groups and products. For (metrizable, complete) nuclear groups one obtains analogues of the Pontryagin duality theorem, of the Bochner theorem and of the Lévy-Steinitz theorem on rearrangement of series (an answer to an old question of S. Ulam). The book is written in the language of functional analysis. The methods used are taken mainly from geometry of numbers, geometry of Banach spaces and topological algebra. The reader is expected only to know the basics of functional analysis and abstract harmonic analysis

Rearrangement of series in nonnuclear spaces

It is proved that if a metrizable locally convex space is not nuclear, then it does not satisfy the Lévy-Steinitz theorem on rearrangement of series