After collecting a number of results on interval and almost interval preserving lin-ear maps and vector lattice homomorphisms, we show that direct systems in variouscategories of normed vector... Show moreAfter collecting a number of results on interval and almost interval preserving lin-ear maps and vector lattice homomorphisms, we show that direct systems in variouscategories of normed vector lattices and Banach lattices have direct limits, and thatthese coincide with direct limits of the systems in naturally associated other cate-gories. For those categories where the general constructions do not work to establishthe existence of general direct limits, we describe the basic structure of those directlimits that do exist. A direct system in the category of Banach lattices and contractivealmost interval preserving vector lattice homomorphisms has a direct limit. Whenthe Banach lattices in the system all have order continuous norms, then so does theBanach lattice in a direct limit. This is used to show that a Banach function space overa locally compact Hausdorff space has an order continuous norm when the topologieson all compact subsets are metrisable and (the images of) the continuous compactlysupported functions are dense. Show less
