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Locally Convex Vector Lattice
In mathematics, specifically in order theory and functional analysis, a locally convex vector lattice (LCVL) is a topological vector lattice that is also a locally convex space. LCVLs are important in the theory of topological vector lattices. Lattice semi-norms The Minkowski functional of a convex, absorbing, and solid set is a called a lattice semi-norm. Equivalently, it is a semi-norm p such that , y, \leq , x, implies p(y) \leq p(x). The topology of a locally convex vector lattice is generated by the family of all continuous lattice semi-norms. Properties Every locally convex vector lattice possesses a neighborhood base at the origin consisting of convex balanced solid absorbing sets. The strong dual of a locally convex vector lattice X is an order complete locally convex vector lattice (under its canonical order) and it is a solid subspace of the order dual of X; moreover, if X is a barreled space then the continuous dual space of X is a band in the order dual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual difference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
