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)
Modular metric spaces. I: Basic concepts. (English) Zbl 1200.54014

The author introduces the notion of a (metric) modular as follows: A (metric) modular on a set X is a function w:(0,)×X×X[0,] satisfying, for all x,y,zX, the following three properties: x=y if and only if w(λ,x,y)=0 for all λ>0; w(λ,x,y)=w(λ,y,x) for all λ>0; and w(λ+μ,x,y)w(λ,x,z)+w(μ,y,z) for all λ,μ>0·

He shows that given x 0 X the set X w ={xX:lim λ w(λ,x,x 0 )=0} is a metric space with metric d w o (x,y)=inf{λ>0:w(λ,x,y)λ}, which he calls a modular space. The article develops the theory of metric spaces generated by (convex) modulars. In this way the author is able to extend results from Nakano’s theory of modular linear spaces to his setting. Among other things, the method can be used to define new metric spaces of (multivalued) functions of bounded generalized variation of a real variable with values in metric semigroups and abstract convex cones.

54E35Metric spaces, metrizability
46A80Modular topological linear spaces