I know I have to use the definition of bounded sequence.
2.
Let be a bounded sequence in a Banach space.
3.
One of the required properties is that a bounded sequence has a cluster point.
4.
A Banach space is reflexive if and only if each bounded sequence in has a weakly convergent subsequence.
5.
Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences ( Alaoglu's theorem ).
6.
In a similar manner, the continuous dual of is naturally identified with ( the space of bounded sequences ).
7.
Now suppose one has a bounded sequence in \ mathbb { R }; by the lemma there exists a monotone subsequence, necessarily bounded.
8.
*If is a uniformly bounded sequence of real valued functions on such that each " f " is Lipschitz continuous with the same Lipschitz constant:
9.
In a Hilbert space, the weak compactness of the unit ball is very often used in the following way : every bounded sequence in has weakly convergent subsequences.
10.
A bounded sequence x with the property, that for every Banach limit \ phi the value \ phi ( x ) is the same, is called almost convergent.
