# 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)
Weak holomorphy and other weak properties. (English) Zbl 1077.46030
In the present short note, the authors show with a very simple proof that, for a vector function with the property that the range of each compact set is contained in an absolutely convex weakly compact set, it suffices to check holomorphy on the elements of a separating subset of the dual of the (locally complete locally convex) range space. That is, the authors have a somewhat stronger hypothesis on the mapping to start with and do not recover Grosse-Erdmann’s result unless the range space is semireflexive. Moreover, contrary to Grosse-Erdmann, the authors also make use of the Dunford-Grothendieck theorem on the equivalence of holomorphy and weak holomorphy.
##### MSC:
 46E40 Spaces of vector- and operator-valued functions 46A20 Duality theory of topological linear spaces 46A32 Spaces of linear operators; topological tensor products; approximation properties 46E10 Topological linear spaces of continuous, differentiable or analytic functions
##### Keywords:
weakly compact set; weak holomorphy; semireflexivity