%0 Journal Article
%A Bombal Gordón, Fernando
%A Vera Boti, Gabriel
%T Means in locally convex spaces and semireflexivity. (Spanish: Medias en espacios localmente convexos y semi-reflexividad).
%D 1973
%@ 0010-0757
%U https://hdl.handle.net/20.500.14352/64738
%X Let S  be a semigroup and let E  be a locally convex topological vector space over the field of real numbers. Let B(S,E)  be the linear space of mappings f:S→E  such that f(S)  is a bounded set of E . For every s∈S  and every f∈B(S,E), s f[f s ]  denotes the function from S  into E  defined by ( s f)(T)=f(s⋅t)  for each t∈S  [(f s )(t)=f(t⋅s)  for each t∈S ]. A subspace X  of B(S,E)  is left [right] invariant if, for every f∈X  and s∈S ,  s f[f s ]  also belongs to X . The space X  is invariant if it is both left and right invariant. The authors give the following definition of a mean: a mean μ  on a subspace X  of B(S,E) , containing the constant functions, is a linear mapping of X  into E  such that μ(f)  belongs to the closed convex hull of f(S) . Moreover. if X  is left [right] invariant, μ  is left [right] invariant provided μ( s f)=μ(f)[μ(f s )=μ(f)] , for every f∈X  and s∈S . If X  is invariant and μ  is left and right invariant, then μ  is invariant.    The authors study the problem of the existence of invariant means on certain subspaces of X . For a larger class of semigroups they prove that if E  is quasicomplete for the Mackey topology, a necessary and sufficient condition to ensure the existence of a invariant mean on B(S,E)  is that E  be semi-reflexive.
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