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(LB)‐spaces of vector‐valued continuous functions
Author(s) -
Frerick Leonhard,
Wengenroth Jochen
Publication year - 2008
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdn033
Subject(s) - mathematics , metrization theorem , compactification (mathematics) , space (punctuation) , function space , pure mathematics , bounded function , discrete mathematics , topology (electrical circuits) , combinatorics , mathematical analysis , separable space , linguistics , philosophy
We consider a question posed by Bierstedt and Schmets as to whether, for an (LB)‐space E and a compact space K , the space C ( K , E ) of E ‐valued continuous functions endowed with the uniform topology is again an (LB)‐space. We present proofs for the case of hilbertizable Montel (LB)‐spaces as well as for the case where E is a weighted space of sequences or functions which even yield that Schwartz's ε‐product Y ε E is an (LB)‐space for every ℒ ∞ ‐space Y . Although the problem for general (LB)‐spaces remains open, we provide several relations and reductions. For instance, it is enough to consider curves, that is, the most natural case K =[0, 1] or, for the class of hilbertizable (LB)‐spaces with metrizable bounded sets, the Stone–Čech compactification of ℕ.