Where in this post we discover an uncanny property of the weak topology: the points of the weak closure of a set cannot always be attained as limits of elements of the set. Naturally, the w-closure of a set is weakly closed. The so-called weak sequential closure of a set, on the other hand, is the set of cluster points of sequences made with elements from that set. The new set is not, however, weakly sequentially closed, which means that there may arise new cluster points; we may, in fact, have to take the weak sequential closure trans-finitely many time to obtain a set which is weakly sequentially closed – still, this may not be weakly closed.
Let be a Banach space. The weak topology in is the coarsest topology in which makes the elements of continuous. In infinitely dimensional spaces, the weak topology is not metrisable, while in finite dimensions the properties of coincide with the ones of as the strong and the weak topology coincide.
A basis of open neighbourhoods of of the weak topology comprises of the sets for some and being a finite collection of elements of
For example, in Hilbert spaces, these sets can be written as
We may use these sets to tell whether a sequence converges weakly to a point ; we need to check whether for every finite collection we have that for every there is a so that for all ,
Alternatively, we may say that converges weakly to if
Instead of testing weak convergence against arbitrary finite collections we may instead use a total set in . Then converges weakly to a point if and only if is -bounded and for every , .
A prime example of a sequence which is weakly convergent, but not -convergent is the sequence of . Since , this is a sequence of elements on the unitary sphere of and so it does not converge strongly. However, for every , the sequence converges to 0, so weakly. Notice that the sequence is of course bounded.
The closure of a set in the strong (norm) topology of is the smallest closed set which contains . This is exactly the same as the set of all (strong) limit points of sequences made with elements from . This is not true in the weak topology the main reason for which being that it is not metrisable. The set of weak sequential cluster points of sequences in is the so-called weak sequential closure of which we will denote by .
This is smaller that the weak closure of :
As an example take the set in . We may show that is closed and weakly sequentially closed, i.e., , but not weakly closed.
Indeed, the origin is not a weak cluster point of : let be a weak basic neighbourhood of the origin. We may, therefore, find elements of in for every and , so is in the weak closure of , but it cannot be attained as a limit of a sequence of elements of .