5 research outputs found
On quantification of weak sequential completeness
We consider several quantities related to weak sequential completeness of a
Banach space and prove some of their properties in general and in -embedded
Banach spaces, improving in particular an inequality of G. Godefroy, N. Kalton
and D. Li. We show some examples witnessing natural limits of our positive
results, in particular, we construct a separable Banach space with the
Schur property that cannot be renormed to have a certain quantitative form of
weak sequential completeness, thus providing a partial answer to a question of
G. Godefroy.Comment: 9 page
Optimal approximate fixed point results in locally convex spaces
Let be a convex subset of a locally convex space. We provide optimal
approximate fixed point results for sequentially continuous maps . First we prove that if is totally bounded, then it has an
approximate fixed point net. Next, it is shown that if is bounded but not
totally bounded, then there is a uniformly continuous map
without approximate fixed point nets. We also exhibit an example of a
sequentially continuous map defined on a compact convex set with no approximate
fixed point sequence. In contrast, it is observed that every affine
(not-necessarily continuous) self-mapping a bounded convex subset of a
topological vector space has an approximate fixed point sequence. Moreover, it
is constructed a affine sequentially continuous map from a compact convex set
into itself without fixed points.Comment: 12 page