Selected topics for the weak topology of Banach spaces

Sun, 20/03/2016 - 12:00

Corson (1961) started a systematic study of certaintopological properties of the weak topology w of Banach spaces E. This
line of research provided more general classes such as reflexive
Banach spaces, Weakly Compactly Generated Banach spaces and the class
of weakly K-analytic and weakly K-countably determined Banach spaces.
On the other hand, various topological properties generalizing
metrizability have been studied intensively by topologists and
analysts. Let us mention, for example, the first countability,
Frechet-Urysohn  property, sequentiality, k-space property, and
countable tightness. Each property (apart the countable tightness)
forces a Banach space E to be finite-dimensional, whenever E with the
weak topology w  is assumed to be a space of the above type. This is a
simple consequence of a theorem of Schluchtermann and Wheeler that an
infinite-dimensional Banach space is never a k-space in the weak
topology. These results show also that the question when a Banach
space endowed with the weak topology is homeomorphic to a certain
fixed model space from the infinite-dimensional topology is very
restrictive and motivated specialists to detect the above  properties
only for some natural classes of subsets of E, e.g., balls or bounded
subsets of E. We collect some classical and recent results of this
type, and characterize those Banach spaces E whose unit ball B_w is
k_R-space or even has the Ascoli property. Some basic concepts from
probability theory and measure theoretic properties of the space l_1
will be used.