# Selected topics for the weak topology of Banach spaces

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.