Weak convergence

From: Julien Santini (santini.julien_at_wanadoo.fr)
Date: 10/30/04 

Date: Sat, 30 Oct 2004 12:23:13 +0200

Hello,

In the following, C denotes the complex numbers.
Let E be a complex Banach space, and A a subset of E. My book calls A
"weakly closed" whenever for any sequence (a_n) of elements of A, if lim
x*(a_n)=x*(a) for any bounded linear functional x*: E->C, then a is in A.

A natural question -that is not mentioned in my book- would be to determine
whether a weakly closed set A is closed for the weak topology, whose
subbasis is given by {x*^(-1)(U); x*: E->C bounded, U open set of C}.

I fail when trying to prove that whenever a is in Adh(A)-A, then there
exists a sequence (a_n) of elements of A weakly converging to a. The fact is
I can control the behaviour of a_n according to a finite number of x* only.
Any suggestion ? - a one word answer would suffice, I'd just want to know if
I should try proving it or not -. 

--
Julien Santini

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