View source for Banach lattice

A vector lattice that is at the same time a [[Banach space|Banach space]] with a norm which satisfies the monotonicity condition:

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A Banach lattice is also called a KB-lineal, whereas an arbitrary normed lattice, i.e. a vector lattice with a monotone norm, is called a KN-lineal. When completing a normed lattice in norm, the order relations may be extended to the resulting Banach space so that it becomes a Banach lattice. If it is possible to introduce in a lattice a Banach topology which converts it to a Banach lattice, such a topology is unique. The simplest example of a Banach lattice is the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b0151602.png" /> of continuous functions on an arbitrary compact topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b0151603.png" /> with the natural (pointwise) order and with the ordinary (uniform) norm. Other examples of Banach lattices include <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b0151604.png" /> spaces and Orlicz spaces (cf. [[Orlicz space|Orlicz space]]). In Banach lattices convergence in norm is (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b0151605.png" />)-convergence for convergence with a regulator. This is not true of normed lattices.

An important special case is a Banach lattice of bounded elements. If a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b0151606.png" /> contains a strong unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b0151607.png" />, i.e. if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b0151608.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b0151609.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516010.png" />, then the smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516011.png" /> for which this inequality is valid is taken as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516012.png" />. The normed lattice thus obtained is called a normed lattice of bounded elements; if it is complete in norm, it is called a Banach lattice of bounded elements. In a Banach lattice (and even in a normed lattice) of bounded elements convergence in norm is identical with convergence with a regulator, while the boundedness of a set of elements in norm is identical with order boundedness. If a normed lattice of bounded elements is conditionally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516013.png" />-complete, it is complete in norm.

The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516014.png" /> is a Banach lattice of bounded elements in which the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516015.png" /> is taken as the unit. For any Banach lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516016.png" /> of bounded elements there exists a compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516018.png" /> is algebraically and lattice isomorphic to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516019.png" />. This is an abstract characterization of the Banach lattice of continuous functions on a compact Hausdorff space.

In any normed lattice an additive functional that is continuous in norm is regular and, moreover, is representable as the difference of two additive functionals which are continuous in norm. In a Banach lattice each positive additive functional is continuous in norm, which means that the classes of regular functionals and additive functionals which are continuous in norm coincide. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516020.png" /> which is dual in the sense of Banach to the normed lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516021.png" /> is a conditionally complete Banach lattice. In a normed lattice the Hahn–Banach theorem may be strengthened as follows: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516022.png" /> there exists a positive additive functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516023.png" />, which is continuous in norm, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516025.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.Z. Vulikh,   "Introduction to the theory of partially ordered spaces" , Wolters-Noordhoff  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.M. Day,   "Normed linear spaces" , Springer  (1958)</TD></TR></table>



====Comments====
Cf. [[Riesz space|Riesz space]] for the notion of  "convergence with a regulator" . The terminology in the article above is from [[#References|[1]]], and is not in general use in the Western literature.

A [[Vector lattice|vector lattice]] (i.e. a real linear (partially) ordered vector space that is at the same time a lattice) is usually called a [[Riesz space|Riesz space]]. A norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516026.png" /> on a Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516027.png" /> is called a Riesz norm if

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whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516029.png" /> (called the monotonicity condition, above). Here

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A normed Riesz space is understood to mean a Riesz space provided with a Riesz norm. The definition of a Banach lattice can now be rephrased as follows: A Banach lattice is a normed Riesz space that is complete in the norm.

All (Riesz) norms on a Banach lattice are equivalent.

For a Banach lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516031.png" /> the order dual space (cf. [[Riesz space|Riesz space]]) and norm dual space coincide (this is not true for normed Riesz spaces in general). A Banach lattice, required to be complete in norm, need not be Dedekind complete. However, any Banach lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516032.png" /> (even, every normed Riesz space) can be regarded as a (Riesz) subspace of a Dedekind-complete Banach lattice (the bi-dual (in norm) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015160/b01516033.png" />, cf. [[Banach space|Banach space]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.H. Schaefer,   "Banach lattices and positive operators" , Springer  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.A.J. Luxemburg,   A.C. Zaanen,   "Riesz spaces" , '''I''' , North-Holland  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Lindenstrauss,   L. Tzafriri,   "Classical Banach spaces" , '''2''' , Springer  (1979)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.C. Zaanen,   "Riesz spaces" , '''II''' , North-Holland  (1983)</TD></TR></table>