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Composition operators on spaces of real analytic functions
Author(s) -
Domański Paweł,
Langenbruch Michael
Publication year - 2003
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.200310053
Subject(s) - mathematics , linear subspace , embedding , locally convex topological vector space , composition (language) , topological space , composition operator , type (biology) , pure mathematics , space (punctuation) , regular polygon , discrete mathematics , algebra over a field , combinatorics , topology (electrical circuits) , banach space , finite rank operator , geometry , ecology , linguistics , philosophy , artificial intelligence , computer science , biology
Let Ω 1 , Ω 2 be open subsets of ℝ d 1and ℝ d 2, respectively, and let A(Ω 1 ) denote the space of real analytic functions on Ω 1 . We prove a Glaeser type theorem by characterizing when a composition operator C φ : A(Ω 1 ) → A(Ω 2 ), C φ ( f ) ≔ f ∘ φ , is a topological embedding. Using this result we characterize when A(Ω 1 ) can be embedded topologically into A(Ω 2 ) as a locally convex space or as a topological algebra. We also characterize LB–subspaces and Fréchet subspaces of A(Ω 1 ). In particular, it follows that if A(Ω 1 ) and A(Ω 2 ) are isomorphic as locally convex spaces, then d 1 = d 2 .