zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Quasi-uniform structures in linear lattices. (English) Zbl 0803.46007
This paper concerns normed linear spaces defined over the field of real numbers. The authors call a normed lattice in which the parallelogram law holds for positive elements an E space, and they call a nonnegative sublinear function q a quasi-norm provided that if q(x)=q(-x)=0 then x=0. Each quasi-normed space has a natural translation-invariant quasi-uniformity 𝒰 q and the functions q * (x)=q(x)+q(-x), q * M(x)=q(x)q(-x), and q * E(x)=(q(x) 2 +q(-x) 2 ) 1/2 define equivalent norms that induce the uniformity 𝒰q * . For the quasi-norm q(x)=x + defined on a normed lattice, the norms defined by q L * , q M , and q E * are equivalent to the original norm and coincide with this norm respectively when the normed lattice is an L space (M space, or E space). Consequently, every normed lattice is determined (in the sense of L. Nachbin) by the quasi-uniformity derived from the quasi-norm q(x)=x + . The authors characterize the positive continuous linear functionals of a normed lattice (E,,) as those linear functionals that are continuous when considered as maps from (E,q) to the real line with the right-ray topology. They use this characterization and the algebraic version of the Hahn-Banach theorem to obtain an alternative proof of Proposition 33.15 from G. J. O. Jameson’s “Topology and normed spaces” (1974; Zbl 0285.46002). This alternative proof illustrates a theme of the paper: It is helpful in determining some global aspects of a normed lattice from the study of its positive cone to decompose the normed lattice into its two quasi-pseudometric structures.

46A40Ordered topological linear spaces, vector lattices
54E15Uniform structures and generalizations
46B40Ordered normed spaces