WEAK AND STRONG CONVERGENCE
DR GAGANDEEP
THEOREM
COROLLARY
If X is reflexive, then weak convergence and weak*convergence in X* are equivalent.
In case, X* is the dual of a non- reflexive space X, then weak convergence and weak*convergence are different. It is preferable to use weak* convergence in X* instead of weak convergence.
THEOREM
PROOF
THEOREM
Let X be a separable Banach space and let M be a bounded subset of X*. Then, every sequence of bounded linear functional in X*
PROOF
Choose a countable dense subset A = { x1 , x2 , …}.
Let { fn} be a sequence in M. then
THEOREM (BANACH-ALAOGLU)
Let X* be the dual of a Banach space X. then, the closed unit ball in X* is weak* compact.