Premium
Uniqueness of the Topology on Spaces of Vector‐Valued Functions
Author(s) -
Villena A. R.
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610701002423
Subject(s) - mathematics , product topology , uniqueness , topology (electrical circuits) , topological vector space , topological group , subspace topology , topological space , general topology , function space , space (punctuation) , banach space , multiplication (music) , pure mathematics , discrete mathematics , mathematical analysis , combinatorics , computer science , operating system
Let Ω be a topological space without isolated points, let E be a topological linear space which is continuously embedded into a product of countably boundedly generated topological linear spaces, and let X be a linear subspace of C (Ω, E ). If a ∈ C (Ω) is not constant on any open subset of Ω and aX ⊂ X , then it is shown that there is at most one F ‐space topology on X that makes the multiplication by a continuous. Furthermore, if U is a subset of C (Ω) which separates strongly the points of Ω and U X ⊂ X , then it is proved that there is at most one F ‐space topology on X that makes the multiplication by a continuous for each a ∈ U. These results are applied to the study of the uniqueness of the F ‐space topology and the continuity of translation invariant operators on the Banach space L 1 ( G , E ) for a noncompact locally compact group G and a Banach space E . Furthermore, the problems of the uniqueness of the F ‐algebra topology and the continuity of epimorphisms and derivations on F ‐algebras and some algebras of vector‐valued functions are considered.