zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Homomorphisms on function lattices. (English) Zbl 1058.54008
Let X be a completely regular space, C(X) the set of all continuous functions from X into the real numbers , and LC(X) a unital vector lattice. The structural space H(L) is the set of all lattice homomorphisms from L into . For each xX the point evaluation map δ x belongs to H(L), where δ x (f)=f(x) for each fL. If L separates points and closed sets, then X is a topolopical subspace of H(L), and H(L) is a realcompactification of X. Indeed, all realcompactifications of X have the form H(L) for some LC(X). In the present paper, the authors study the realcompactifications H(L) and compactifications H(L * ) of X, where L * denotes the set of bounded elements in L, and H(C * (X)) is the Stone-Čech compactification βX of X. They call X L-realcompact in case H(L)=X. They show, among other things, that for each ϕH(L) there exists ξβX such that, for all fL, f β (ξ) and ϕ(f)=f β (ξ). They study the case where L=Lip(X) is the set of real valued Lipschitz functions on a metric space X and show that X is L-realcompact if, and only if, every closed ball in X is compact. Other results concern a vector lattice isomorphism T between Lip(X) and Lip(Y) which maps unit element to unit element. They show that if X,Y are complete, then T exists if, and only if, X and Y are Lipschitz homeomorphic. A similar result for Lip * (X) and Lip * (Y) was obtained by N. Weaver [Pac. J. Math. 164, No. 1, 179–193 (1994; Zbl 0797.46007)].
MSC:
54C40Algebraic properties of function spaces (general topology)
46E05Lattices of continuous, differentiable or analytic functions
54D35Extensions of topological spaces (compactifications, supercompactifications, completions, etc.)
54D60Realcompactness and real compactification (general topology)