Normed Topological Pseudovector Groups

Abstract

A normed topological pseudovector group (NTPVG for short) is a valued topological group (V, +, · ) (not necessarily Abelian) endowed with a continuous scalar multiplicationt · x = t x for each t, s ∈ R + and x, y ∈ V. It is shown that every valued topological group can be isometrically and group-homomorphically embedded in a NTPVG as a closed subset by means of a functor. Locally compact NTPV groups are fully classified. It is shown that the (unbounded) Urysohn universal metric space can be endowed with a structure of a NTPV group of exponent 2.Keywords Normed vector spaces · Topological groups · Locally compact groups · Valued groups · Dynamical systems · Urysohn universal metric space Mathematics Subject Classifications (2010) 22A99 · 46B99 · 46A99Normed vector spaces are metrizable topological groups of a very special kind. One may say that these are groups 'modelled' on the additive group of real numbers. The possibility of multiplying elements of a group by reals in a homomorphic way and the convex structure of a group facilitate investigations of such groups and has many deep topological and geometric consequences. For example, every normed vector space is an absolute extensor for metric spaces ([5]; see also [2, Theorem II.3.1]), locally compact normed vector spaces are finite dimensional and are uniquely determined (up to topological group isomorphism) by their dimension, non-locally compact completely metrizable normed spaces (that is, infinitedimensional Banach spaces [9]) as topological spaces are also uniquely determined P. Niemiec (B)

Universal valued Abelian groups

“…The part on pseudovector structures extends our earlier (introductory) work [18] in this subject. The proofs presented here are quite new and much more general.…”

“…The proofs presented here are quite new and much more general. Theorem 1.3 generalizes and strengthens Theorem 4.3 of [18]. Since every norm on a nontrivial pseudovector group is unbounded, to equip the groups G 1 (N) with 'normed-like' pseudovector structures we have to extend the notion of a norm to a subnorm, which is done in the recent paper.…”

“…Pseudovector groups were introduced in [18]. Here we shall generalize the results of [18] on pseudovector structures on the (unbounded) Boolean Urysohn group, mainly in order to prove that all the groups G r (N)'s are homeomorphic to the Hilbert space.…”

“…) is satisfied, thanks to the isometricity of ϕ 1 . Otherwise, follows from (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18). The case of j = 2 is similar and is left for the reader.…”

“…In Section 8 we prove that each of the topological spaces G r (N) is homeomorphic to the Hilbert space l 2 . To establish this result we develop our earlier study of the so-called topological pseudovector groups, introduced in [18]. Namely, a pseudovector Abelian group is a triple (G, +, * ) such that (G, +) is an Abelian group, ' * ' is an action of [0, ∞) on G satisfying the following axioms: 0 * x = 0 G , 1 * x = x, (st) * x = s * (t * x) for all x ∈ G and s, t 0, and for every t 0 the function G ∋ x → t * x ∈ G is a group homomorphism.…”