The Properties Of Direct-Sums Of Banach Lattices

The decomposition (including directed sum) of linear spaces、Riesz spaces (Banach lattices、normed Riesz spaces (Banach spaces) is the important aspect of the studying on spatial structure properties. Such as the n dimensional Euclidean space is the directed sum of n real number.The classical sequence space lp can be seen as the lp -directed sum of countable real spaces. And a number of important spatial structure properties meet different forms of directed sum decomposition. So the main work of this paper is to discuss the lp-directed sum about Banach spaces. And it mainly focuses on the global properties of spatial directed sum. That is the link between directed sum and coordinate spaces. It also discusses the subset properties in spatial directed sum.The paper first introduces a series of concepts about Banach lattices’directed sum and gives the relatively normed definition. It also considers the Dedekind completely of (?)lpEi and (?) co Ei are considered, and obtains these spaces are complete if and only if every Ei is complete. Similarly, the necessary and sufficient conditions of the Dedekind completeness of lp (E) and c0(E) are obtained.Secondly, this paper studies the KB-properties、reflexivity and conjugacy of lp-directed sum about Banach lattices, and mainly obtaines the following several conclusions:for 1≤p<+∞,(?)lp Ei is KB-space if and only if every Ei is a KB-space; (?) lp Ei is AL-space if and only if every Ei is a AL-space; (?) lp Ei is AM-space if and only if every Ei is a AM-space.At last, it discusses the subset properties of lp-directed sum about Banach lattices. That is, for 1≤p<+∞,A (?) (?) lp Ei is relatively compact if and only if every Pi A in Ei is relatively compact and lim sup‖Pnx‖=0...