Finite Elements in Vector Lattices

Mar tin R. Web er ( Dr es d en) Abstract. In vector lattices of continuous functions on a locally com p act H ausdorff space a natural and im portant role play the fi nite functions, i. e. continuous fu nctions w ith a com pact support. F inite and totally fi nite elem ents are the abstract notion of such functions in arbitrary A rchim edean vector lattices. T his survey describes the m ain properties of fi nit e and totally fi nite elem ents in arbitrary and norm ed vector lattices and in their sublattices. For m ost of the classical vector lattices the collection of all fi nite elem ents is indicated. F inite and totally fi nite elem ents and their relations to fi nite functions are studied in vector lattices of continuous functions. In spec ial vector lattices of regular operators the fi rst results on their fi nite elem ents w ill be presented. T hen ther e is investigated the interesting question on the behaviour of fi nite elem ents, w hen the vector lattice is repr esented (isom orphically) as a vector lattice of continuous functions on a locally com pact space, w here one e xpects that fi nite elem ents are m apped into fi nite functions. S pecial situations are discussed w here th is is the case. B y m aking use of the space of m axim al ideals, in particular, a topological characteriza tion of fi nite and totally fi nite elem ents by m eans of com pact subsets is given