27 research outputs found
Uncountable sets of unit vectors that are separated by more than 1
Let be a Banach space. We study the circumstances under which there
exists an uncountable set of unit vectors such that
for distinct . We prove that such a set exists
if is quasi-reflexive and non-separable; if is additionally
super-reflexive then one can have for some
that depends only on . If is a non-metrisable compact,
Hausdorff space, then the unit sphere of also contains such a subset;
if moreover is perfectly normal, then one can find such a set with
cardinality equal to the density of ; this solves a problem left open by S.
K. Mercourakis and G. Vassiliadis.Comment: to appear in Studia Mat
The ideal of weakly compactly generated operators acting on a Banach space
We call a bounded linear operator acting between Banach spaces weakly
compactly generated ( for short) if its range is contained in a
weakly compactly generated subspace of its codomain. This notion simultaneously
generalises being weakly compact and having separable range. In a comprehensive
study of the class of operators, we prove that it forms a closed
surjective operator ideal and investigate its relations to other classical
operator ideals. By considering the th long James space
, we show how properties of the ideal of
operators (such as being the unique maximal ideal) may be used
to derive results outside ideal theory. For instance, we identify the
-group of as the additive group of
integers
On a composite functional equation fulfilled by modulus of an additive function
We deal with the problem of determining general solutions f : R → R of the
following composite functional equation introduced by Fechner:
f(f(x) − f(y)) = f(x + y) + f(x − y) − f(x) − f(y).
Our result gives a partial answer to this problem under some assumptions upon f(R). We
are applying a theorem of Simon and Volkmann concerning a certain characterization of
modulus of an additive function. A new proof of their result is also presented