Concepts in vector spaces with convergence structures

G. F. C. de Bruyn
1975 Canadian mathematical bulletin  
1. Introduction. Limit vector spaces (iimitierte Vektorraume') were defined by Fischer [1] and concepts such as continuity, compactness, etc. were introduced and studied by him and others, e.g. by Kent [3]. In this paper the concepts of precompactness and boundedness in limit vector spaces are studied. It is shown that most of their properties in topological vector spaces hold in limit vector spaces. The notions of 'espaces à bornés' (tfè-spaces) and 'espaces à bornés complets' (tfèc-spaces)
more » ... s' (tfèc-spaces) were introduced by Waelbroeck [4]. An aZ?-space (E, 3$) is a vector space E over the field of complex numbers in which a class 3% of subsets of E is specified such that each finite subset of E is an element of ^, the union and sum of two elements of 3$ are elements of 3 §, the convex envelope, any scalar multiple and subset of an element of 3$ are elements of 3%, and no non-zero vector subspace of E is an element of 3 §. An a&ospace is an aè-space having, together with some other properties, the property that the absolutely convex envelope of an element of 3$ is an element of 3$. It is shown that under certain conditions a limit vector space, with 3$ the class of bounded subsets, is an aô-space and vice versa.
doi:10.4153/cmb-1975-091-3 fatcat:qysk5awjlzgabeyrjevb6dihku