Applied Analysis Seminar

**Monday June 3
in K9509, SFU
**

10:30 - 11:20
(note time)

Vaclaz Zizler, Edmonton

**Abstract:**
A Markushevich basis {x_{a},f_{a}}_{a Î G} for
a Banach space X is a biorthogonal system in X such that
[` span ]{x_{a}}_{a Î G} = X and
{f_{a}}_{a Î G} is separating points in X.
We will call such a basis a s- shrinking Markushevich basis if
for every e > 0 the set G can be split into
G = È_{n = 1}^{¥}G_{n}^{e} such that
(G_{n}^{e})¢ Ì B_{X**}^{e},
for each n Î IN, where
B_{X**}^{e}
is the e- ball in X^{**} and (G^{e}_{n})¢ is the set of all accummulation points
of
{x_{a}}_{a Î Gen} in X^{**}.
We will give a sketch of a proof to the fact that
a Banach space X admits a s-
shrinking Markushevich basis if and
only if X is a subspace of a weakly compactly generated space
Z i.e. such a space Z that there is a weakly compact set K in Z whose span is dense in Z.
If this happens for X, we will sketch a proof that then any normalized Markushevich basis for X
is s- shrinking. We will show how this approach yields
a short proof to the known result of Y. Benyamini,
M. E. Rudin and M. Wage, namely
that a continuous image of an Eberlein compact is an
Eberlein compact.
The compact space K is an Eberlein compact if K is
homeomorphic to a weakly compact set in a Banach space
(in its weak topology).
We will discuss several open problems in this area.

The talk will be based on a recent joint work of M. Fabian, V. Montesinos and V. Zizler.

File translated from T

On 19 May 2002, 13:18.