## Additive Subgroups of Topological Vector SpacesThe 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. |

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abelian group abelian topological group arbitrary assume Banach space base at zero Borel measure bounded subset canonical projection Choose closed subgroup compact subset continuous characters continuous unitary representation conv convergent series defined denote dense direct sum duality dually closed dually embedded ellipsoid equicontinuous exists finite subset follows formula g e G group G Hence Hilbert space homomorphism implies infinite dimensional integer LCA groups Lemma Let G Let H linear functional linear operator linear subspace locally convex space locally quasi-convex group Math metrizable metrizable and complete n-dimensional n=l n non-trivial normed spaces NQ(G nuclear group nuclear spaces nuclear vector group numbers orthogonal pre-Hilbert seminorms precompact Proof prove quotient group Radon measure real lines reflexive group sequence span strongly reflexive subgroup of G subset of G Suppose theorem topological isomorphism topological vector space topology on G unitary representation voln