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The Approximation Property for Nuclear Convergence Vector Spaces
Author(s) -
Bjon Sten
Publication year - 1989
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19891420118
Subject(s) - mathematics , countable set , convergence (economics) , holomorphic function , zero (linguistics) , pure mathematics , vector space , modes of convergence (annotated index) , identity (music) , locally convex topological vector space , space (punctuation) , compact convergence , mathematical analysis , topological vector space , rate of convergence , topological space , isolated point , linguistics , philosophy , physics , channel (broadcasting) , engineering , acoustics , electrical engineering , economics , economic growth
Nuclear convergence spaces are studied. It is shown that an L e ‐embedded convergence vector space E is L e L M ‐embedded if it is Schwartz and satisfies a certain countability condition which expresses that the set of filters converging to zero is essentially countable. Further it is shown that if E is L e L M ‐embedded and nuclear, then the identity E → E can be approximated with finite operators in the equable continuous convergence structure on L(E, E). This result is used in the study of the spectrum Hom c H e ( U ) of the convergence algebra H e ( U ) of holomorphic functions on a circled convex open set to prove sufficient conditions for the validity of the formula Hom c H e ( U ) ∼ U .