Direct limits in categories of normed vector lattices and Banach lattices

Abstract After collecting a number of results on interval and almost preserving linear maps vector lattice homomorphisms, we show that direct systems in various categories normed lattices Banach have limits, these coincide with limits the naturally associated other categories. For those where general constructions do not work to establish existence describe basic structure exist. A system category contractive homomorphisms has limit. When all order continuous norms, then so does This is used function space over locally compact Hausdorff an norm when topologies subsets are metrisable (the images of) compactly supported functions dense.